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THOMPSON ,  BROWN  &  C  O 


Digitized  by  the  Internet  Archive 

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§rabburij's  glat^emalical  Sm^s. 


ALGEBRA  FOR  BEGINNERS, 


BY 

WILLIAM     F.    BRADBURY,    A.M. 

I  r 

HEAD  MASTER  IN  THE  CAMBRIDGE  LATIN  SCHOOL, 
AND 

GRENVILLE    C.    EMERY,   A.M., 

MASTER   IN  THE   BOSTON   LATIN    .SCHOOL 


BOSTON: 
THOMPSON,  BROWN,  AND    COMPANY. 

23  Haw  LEY  Street. 


Copyright,  1894, 
By  William  F.  Bradbvky  and  Grenville  C.  Emery. 


CAJORI 


John  Wilson  anu  Son,  Cambridge,   U.S.A. 


PREFACE. 


This  work  has  been  prepared  expressly  for  beginners,  and 
in  response  to  a  call  for  an  Algebra  for  the  higher  classes 
in  Grammar  Schools. 

It  is  taken  for  granted  that  the  pupil  is  familiar  with  the 
principles  of  ordinary  Arithmetic. 

Few  rules  and  detinitions  are  given ;  the  use  of  algebraic 
language  is  illustrated  by  numerous  exercises  ;  and  the  elemen- 
tary principles  of  Algebra  are  made  clear  by  the  introduction 
of  easy  problems. 

In  order  to  awaken  the  interest  of  the  pupil,  the  Equation, 
its  reduction,  and  numerous  problems  are  introduced  at  the 
very  beginning  of  the  book. 

The  examples  given  are  carefully  graded  from  the  very 
simple  to  the  more  difficult.  A  considerable  part  of  them 
have  been  tested  by  teachers  of  the  grade  for  which  they  are 
designed.  After  the  earlier  problems,  care  has  been  taken  to 
introduce  for  the  most  part  such  as  require  algebraic  princi- 
ples in  their  solution  rather  than  those  that  can  be  more 
easily  and  naturally  solved  without  the  aid  of  Algebra. 
There  is  nothing  better  for  training  the  intellectual  powers 
than  putting  into  algebraic  language  the  conditions  of  a 
mathematical  problem. 

911387 


IV  PREFACE. 

The  subjects  introduced  and  the  method  of  treatment  are 
such  as  to  give  the  pupil  a  substantial  ground-work  for  the 
more  advanced  work  in  Algebra. 

It  is  not  essential  that  every  example  and  problem  given 
in  any  topic  should  be  done  before  the  pupil  passes  to  the 
succeeding  subject.  The  teacher  should  use  his  judgment  in 
each  case,  varying  the  number  of  exercises  taken  according 
to  the  ability  of  the  pupil  and  the  time  allowed  for  the  study. 
This  is  especially  true  in  the  subject  of  Factoring,  of  the 
Greatest  Common  Divisor  of  Polynomials,  and  of  the  Least 
Common  Multiple  of  Polynomials.  Further,  it  is  not  essen- 
tial to  follow  in  every  respect  the  order  of  topics  given  in  the 
book.  If  the  teacher  does  not  approve  of  the  plan,  adopted 
by  the  authors,  of  beginning  with  the  Equation,  he  can  defer 
the  whole  or  any  part  of  the  first  four  chapters  till  a  part  or 
the  whole  of  Chapters  V.-XIV.  is  completed. 

Some  of  the  miscellaneous  equations  and  problems  at  the 
end  of  the  book  are  a  little  more  difficult  than  those  in  the 
body  of  the  work,  and  are  to  be  selected  for  use  at  the  dis- 
cretion of  the  teacher. 

W.  F.  B. 
G.  C.  E. 

Cambridge,   Mass.,   March,    1894. 


TABLE   OF   CONTENTS. 


CHAPTER  I. 

Page 

Introduction 1 

Signs.     The  Equation 2 

Axioms      . 3 

Exercises 4 

Problems 6 

CHAPTER   n. 

Reduction  of  Simple  Equations 9 

CHAPTER   HI. 

Oral  Exercises 15 

Written  Exercises 16 

Problems 19 

CHAPTER   IV. 

Equations  containing  Two  or  more  Unknown  Numbers  29 

Problems 36 

CHAPTER  V. 

Algebraic  Numbers      .     .          41 

CHAPTER   VI. 

Addition       43 


VI  TABLE   OF   CUNTENTS. 

CHAPTER    VII. 

Page 

Subtraction 53 

CHAPTER  VIII. 
Multiplication 59 

CHAPTER  IX. 
Division 66 

CHAPTER   X. 
Theorems  ok  Development 73 

CHAPTER   XI. 
Factoring     .  78 

CHAPTER   XII. 
Greatest  Common  Divisor 92 

CHAPTER  XIII. 
Least  Common  Multiple 97 

CHAPTER   XIV. 
Fractions 100 

CHAPTER    XV. 
Generalization ll-l 

CHAPTER   XVI. 
Miscellaneous  Examples 119 


ALGEBRA   FOR  BEGINNERS. 


CHAl'TEE    I. 

(See  Pkeface.) 

1.  Mr.  Hardy  had  three  sons,  Francis,  Ralph,  and  John. 
On  one  occasion  he  distributed  among  them  some  marbles, 
giving  to  Francis,  the  eldest,  twice  as  many  as  to  Ealph, 
and  to  Ralph  twice  as  many  as  to  John,  the  youngest. 

If  now  we  are  told  that  he  gave  John  2  marbles,  we  at^ 
once  know  that  Ralph  received  4,  and  Francis  8. 

Suppose  now  we  let  n  stand  for  tlie_ntimb£rjQ£,  jiiarbles_ 
jTohn_  receives :  then  2  n  will  stand  for  the  number  Ralph 
receives;  and  4n  for  the  number  Francis  receives;  and  n 
and  2  n  and  4  n,  or  7  n,  will  be  the  wliole  number  of  mar- 
bles given  to  the  three  boys. 

n  and  2  n  and  4  ii  equal  7  n. 

For  and  we  generally  use  in  Algebra  plus. 

n  plus  2  n  plus  4  n  equals  7  n. 

2.  About  the  year  1550  a.  D.,  Stifel,  a  German,  suggested 
the  use  of  the  sign  +  for  the  word  plus ;  and  shortly  after 
this,  Robert  Recorde,  an  Englishman,  invented  the  sign  =, 
for  equals. 

For  the  statement  above,  then,  we  write 

n  +  2  n  +  4  n  =  7  n 
1 


2  ALGEBRA. 

3.  It  is  easily  seen  that  the  nvimber  of  marbles  the  father 
gives,  less  the  number  he  gives  the  eldest  son  Francis,  is 
equal  to  the  number  he  gives  to  lialph  and  John ;  or 

7  iv  less  4  It  =  '1  n  +  ih 

n    For-/''-«5.%VG  generally  use  in  Algebra  minus. 

7  n  minus  4  n  =  '2  ii  4-   ii 

miiiu 
—  ,  and  we  now  write 

7  )i  —  4:  n  —  2  n  +  n 

4,  For  the  sake  uf  l)revity,  signs  were  invented  for  use 
not  only  in  addition  and  subtraction,  but  also  in  multipli- 
cation and  division. 

Thus  for  ■maltiplied  hi/  we  use  the  sign  X,  or  the  dot  • 
above  the  line ;  thus,  7  X  5,  or  7  •  5,  means  that  7  is  to  be 
multiplied  by  5.  Between  a  figure  and  a  letter,  and  between 
letters,  the  sign  of  multiplication  is  usually  omitted  ;  thus  7  n 
means  7  X  it.  But  when  figures  are  to  be  multiplied  together 
the  sign  (if  multiplication  cannot  be  omitted ;  thus  35  does 
not  mean  8  X  5  but  3  tens  and  5  units,  or  thirty-five.  For 
divided  hij  we  use  'the  sign  -^ ,  or  : ;  thus,  8  -^  4,  or  8  :  4, 
means  tliat  8  is  to  be  divided  by  4.  Division  is  also  ex- 
]iresse(l  by  the  fractional  form  |.  All  these  expressions  for 
division  liave  for  their  value  2. 

6.  The  sign  .".is  often  used  instead  of  the  word  hence,  or 
ihcrefiyrr. 

6.  The  expression  n  -\-  1  n  +  4  n  =  7  n  is  called  an  equa- 
tion. The  parts  to  the  left  and  right  of  the  sign  =  are 
called  mcmhers,  or  sides,  and  are  distinguished  as  the  first 
viemhcr  and  second  inemher,  or  the  left  side  and  right  side. 


ALGEBRA. 


7.  An  equation,  and  tlie  processes  used  in  its  reduction, 
can  be  illustrated  by  a  lever,  or  see-saw,  with  arms  of  equal 
length  ;  the  fulcrum,  or  balancing  point,  representing  the  sign 
(^—)  uf  equality,  and  the  arms  the  sides  of  the  equation. 


^"ijfe^^^^fe 


Arthur  +  Bertha  =  Caroline  +  David 
A  +  B  -  C  +1) 

8.  It  is  evident  in  order  that  the  lever  may  remain  in  its 
horizontal  position  tliat  Arthur  and  Bertha  must  together 
weigh  the  same  as  Caroline  and  David;  and  if  the  weight  on 
either  end  is  changed  in  any  way,  the  weight  on  the  other 
end  must  be  changed  in  like  amount.  So,  in,  an  equaf/iov, 
whatever  is  done  to  one  side  must  he  done  to  the  other,  in  order 
that  the  equality  may  remain.     That  is, 

1.  If  anythhig  is  added  to  one  side,  an  equal  amount 
must  be  added  to  the  other. 

2.  If  anything  is  subtracted  from  one  side,  an  equal 
amount  must  be  subtracted  from  the  other. 

3.  If  one  side  is  multiplied  by  any  number,  the  other 
side  must  be  multiplied  by  an  equal  number. 

4.  If  one  side  is  divided  by  any  number,  the  other  side 
must  be  divided  by  an  equal  number. 

Such  self-evident  statements  are  called  Axioms. 


4  ALGKBRA. 

9.     Exercises  on  the  Application  of  the  Axioms. 

I.  A  man  received  for  his  day's  wages  12  and  tlien  spent  $2. 
0  How  much  money  out  of  his  day's  wages  did  he  have  left  ? 

/  r      2.    A  boy   who  had  10  marbles  lost  6  ;    then  he  bought  6. 
How  many  did  he  have  then  ? 

3.  y  —  6  +  6  =  ?     ^-  +  8  -  8  z=  ?     z  —  a^a^l 

4.  If  5  a-  =  15,    what   does   5  a;  +  7   equal  ?     What    axiom 
applies  ? 

5.  If  a:  —  3  =  10,   how  can  we  find  the  value  of  x  ?     What 
axiom  applies  ?     What  does  x  equal  "/ 

().    Suppose  X  =  4,  how  can  we  change  the  equation  so  that 
the  right  side  shall  be  5? 

7.  If  X  +  8  =  ]7,  what  does  x  +  12  ecjual  ?     x  +  4  ?     What 
does  X  equal  ? 

8.  If  cc  =  3,  what  does  8  x  e<iual  ?     What  axiom  ? 

9.  If  a;  =  5,  what  does  4  a;  +  7  (Mjual  ?     -   / 

10.  If  f  =  7,  what  does  X  e<]ual  ?     What  axiom  ?       ^  / 

o 

II.  If  "^^"^  =  20,  what  does  4  ./•  equal  ? 

12.    If  5  a:  =  30,  what  does  a;  equal  ?     What  axiom  ? 

/^  3  a:  3  a: 

^     14.    If    ^    —  8  =  10,  liow   can    we   get   the  value  of   -—  ? 
5  5 

What  axiom  ?     How  then  can  we  get  the  value  of  3  a:?     How 

then  the  value  of  a;  ? 

/"^  3  35  X 

'      15.    How  from  -p-  =  18  can  we  get  the  value  of    '^  ?     How 
then  the  value  of  a;  ? 

10   Express  in  the  form  of  equations  the  following  statements. 

1.  c  is  equal  to  a  added  to  h. 

2.  Thirteen  exceeds  seven  by  six. 

3.  The  excess  of  9  over  5  is  equal  to  4. 


13.   Find  ^  if  ^  =  15.     Find  a:. 


f'^- 


t 


EXERCISES.  5 

4.  The  excess  of  a  over  h  is  equal  to  c. 

5.  c  is  3  less  than  a  +  f>- 

6.  a  is  as  uiuch  greater  than  5  as  oO  is  greater  than  c. 

7.  a  is  four  times  as  great  as  b. 

8.  Three  times  a  exceeds  29  by  b. 

\  9.    Four  times  //  is  equal  to  the  excess  of  x  over  2. 
10.    a  subtracted  from  a  is  equal  to  zero. 

11.     Exercises  in  the  Use  of  Algebraic  Language. 

1.  If  a  pear  cost  x  cents  and  an  orange  2  x  cents,  what  will 
represent  the  cost  of  both  ?  Ans.    x  cents  -\-  2  x  cents. 

2.  If  John  has  marbles  to  the  number  of  3  x,  and  Henry  to 
the  number  of  5  x,  how  many  marbles  do  they  have  together  ? 

3.  If  |2,  $3,  and  14,  are  together  $9,  what  are  2  x,  3  x, 
and  4  x  together  ? 

4.  If  A  earns  $2  x,  B  $o  x,  C  $4  x,  in  a  day,  how  many 
dollars  do  A,  B,  and  C  together  earn  in  a  day  ? 

5.  If  X  represent  A's  money  in  dollars,  and  B  has  twice  as 
much  as  A,  and  C  three  times  as  much  as  B,  what  will  repre- 
sent the  entire  amount  of  their  money  ? 

6.  If  an  apple  cost  x  cents,  what  will  represent  the  cost  of 
2  apples  ?     Of  3  apples  ?     Of  4  ?     Of  5  ? 

7.  If  A  can  travel  y  miles  in  an  hour,  how  many  miles  can 
he  travel  in  2  hours  ?     In  5  hours  ?     In  7  ?     In  9  ? 

8.  If  a  contractor  pays  x  dollars  a  day  to  each  of  his  men, 
what  does  he  pay  to  each  man  for  6  days'  work  ?  What  does  he 
pay  6  men  for  a  day's  work  ?  To  3  men  for  2  days'  work  ?  To 
2  men  for  3  days'  work  ? 

9.  What  will  12  yards  of  cloth  cost  at  f  2  a;  a  yard  ?  8  yards 
at  $3  cc  a  yard  ?     3  yards  at  $8  a;  a  yard  ? 

10.  If  3  bags  of  coffee  cost  3  x  dollars,  what  does  one  bag  of 
cost  ?     5  basrs  ? 


11.    li  8x  dollars  is  the  price  of  4  bushels  of  corn,  what  is 
that  a  bushel  ? 


s 


b  ALGEBRA 

12.  What  must  I  pay  for  10  barrels  of  flour  if  I  can  buy 
8  barrels  for  24  x  dollars  ? 

13.  ^Ir.  Wallace,  \\li<>  is  S  ./•  yeai's  old,  lias  a  son  Arthur 
half  his  age.  Arthur  has  a  daughter  Alice  also  half  of  his 
age.      How  old  is  Alice  ? 

14.  If  8  bushels  of  corn  are  worth  Ki  x  dollars,  and  5  barrels 
of  apples  15  x  dollars,  how  many  bushels  of  corn  ought  to  be 
given  for  6  barrels  of  apples  ? 

12.     PROBLEMS. 

1.  Two  boys,  James  and  Henry,  had  together  50  cents,  and 
Henry  had  4  times  as  many  as  James,  how  many  did  each 
have  ? 

If  now  we  knew  how  many  James  had,  we  could  find  the  number 
Henry  had.  Suppose  we  represent  this  now  unknown  number  by  o:. 
Then  x  will  represent  the  number  of  cents  James  had,  and  4  times  a;, 
or  4  a;,  the  nimiber  of  cents  Henry  had,  and  x-\-  A.x  the  number  of  cents 
both  together  had. 

Then  a;  +  4  ,i:  =  50 

or  5  X  =  .5(» 

and  X  —  10,  the  no.  of  cents  James  had. 

4  ,,•  =  40,         "  "       Henry    " 

2.  Three  men,  A,  B,  and  C,  form  a  company  with  a  capital 
of  $3000.  C  put  in  three  times  as  much  as  A,  and  B  twice  as 
much  as  A.      How  many  dollars  did  each  put  in  ? 

Let  :i;  =  the  no.  of  dollars  A  put  in, 

then  2  a;  =       "  "  B      '• 

and  3:0=       "  "  C      " 

and  a;  +  2  a;  +  3  a;  =       "  "  they  all  put  in. 

But  they  all  put  in  $3000  ; 
.'.  x  +  2x  +  3x  =  3000 

6  a;  =  3000  ► 

x=    500 
2  a;  =  1000 
3.r=:L50O 
Therefore,  A  put  in  $500,  V,  .$1000,  and  V  %  1500. 


PROBLEMS,  7 

3.  The  sum  of  two  numbers  is  84,  and  the  greater  is  twice 
the  less.     What  are  the  numbers  ?  Ans.     28  and  56. 

4.  Two  men,  A  and  B,  contribute  to  a  fund  $40,  of  which 
B  gives  '$10  more  than  A.      What  does  eacli  give  ? 

Let  X  —  the  no.  of  dollars  A  gives, 

then  X  +  10  =       "  "  B     " 

and  X  +  X  -\-  \0  =       "  *'  A  and  B  give. 

But  A  and  B  together  give  $40. 

X  +  a;  +  10  =  40 
or  2  X  +  10  =  40 

Now  10  =  10 

By  subtraction  2  a;  =  30  (Ax.  2.) 

X  =  15,  the  no.  of  dollars  A  gives, 
and  a; +10  =  25,       "  "  B     " 

5.  The  number  of  ^^ears  in  the  ages  of  two  boys,  Dick  and 
Jack,  added  together  make  31,  and  Dick  is  five  years  older 
than  Jack.     What  are  the  ages  of  Dick  and  Jack  ? 

Let  X  —  the  no.  of  years  in  Dick's  age, 

then  X  —  5  =       "           "              Jack's  " 

and  X  +  a;  -  5  =  31                 , 

or  2x-5  =  31                  ' 

But  5=5 

By  addition  2  x  =  30                                                  (Ax.  1.) 

X  =  18,  the  no.  of  years  in  Dick's  age, 

and  X  —  5  =  13,        "             "           Jack's    " 

6.  In  a  school  there  are  16.5  pupils  and  twice  as  many  boys 
as  girls.     How  many  boys  are  there  ? 

7.  Four  times  a  certain  number  added  to  three  times  the 
same  number  gives  56.     Find  the  number. 

8.  A  farmer  being  asked  how  many  sheep  he  had,  said  that 
if  he  had  5  timfes  as  many  more  he  should  have  240.  How 
many  had  he  ? 

9.  The  sum  of  two  numbers  is  .344,  and  the  greater  is  7  times 
the  less.      What  are  the  two  numbers  ? 


8  ALGEBRA. 

10.  A  horse  and  carriage  are  together  worth  -f  320,  and  the 
horse  is  worth  $100  more  than  the  carriage.  What  is  eacli 
worth  ? 

11.  A  farmer  has  a  horse,  a  cow,  and  a  sheep.  Tlie  horse 
is  worth  3  times  as  much  as  the  cow,  and  the  cow  8  times  as 
much  as  the  slieep,  and  all  together  are  worth  $198.  How 
much  IS  each  worth  ?    Ans.   Sheep,  $6  ;  cow,  f  48  ;  horse,  $144. 

The  answers  to  examples  should  be  vcrijied,  that  is,  shown 
to  fulfil  the  given  conditions.     Thus,  in  Prob.  11,  it  is  stated, 

(1)  That  the  horse  is  worth  3  times  as  much  as  the  cow, 

(2)  That  the  cow  is  worth  8  times  as  much  as  the  sheep,  and 

(3)  That  all  together  are  worth  $198.     These  conditions  are 
all  fulfilled  by  the  prices  $144,  $48,  and  $6,  respectively. 

12.  Two  partners,  A  and  B,  have  in  the  firm  property  to  the 
value  of  $6546,  and  B  has  $500  more  than  twice  as  much  as  A. 
What  is  the  share  of  each  ?    )\ 

13.  The  sum  of  the  number  of  years  in  the  ages  of  a  father 
and  his  son  is  70,  and  2  years  ago  the  father's  age  was  twice 
the  son's.    What  is  the  age  of  each  ?     Ans.   Son,  24;  father,  46. 

14.  Three  orchards  bore  48  barrels  of  apples,  of  which  the 
first  bore  twice  as  many  as  the  second,  and  the  third  as  many 
as  the  other  two.      How  man}^  barrels  did  each  bear  ? 

15.  Three  men.  A,  B,  and  C,  received  $138  for  digging  a 
ditch.  A  dug  3  rods  while  B  was  digging  2  and  C  1.  How 
much  should  each  receive  ? 

16.  A  man  is  three  times  as  old  as  his/ son,  and  his  .son  twice 
as  old  as  his  daughter,  and  the  sum  ot  the  number  of  years  of 
their  ages  is  50.     AYhat  is  the  age  of  each  ? 

17.  A,  B,  and  C  hired  a  bouse  for  $500.  of  wliicb  A  was  to 
pay  twice  as  much  as  B.  and  B  $60  less  than  C.  How  much 
was  each  to  pay  ? 

18.  Two  flocks  of  shee])  are  equal  in  number,  but  if  45  are 
transferred  from  one  to  the  other,  one  will  have  four  times  as 
many  as  the  other.     How  many  were  there  in  the  original  flocks? 


REDUCTION   OF   SIMPLE   EQUATIONS. 


CHAPTER    II. 

13.     REDUCTION    OF    SIMPLE    EQUATIONS. 

Find  the  value  of  x  iu  the  following  equations  : 
1.   5  X  -  9  =  3  a;  +  3  (1) 

If  we  add  9  to  both  sides  of  (1)  we  have, 

5  X  -  9  +  9  r=  3  X  +3  +  9  (2) 

or  5  X  =  3  X  +  3  +  9  (3) 

If  we  subtract  3  x  from  both  sides  of  (3)  we  have, 

5x-3x  =  3a:-3x  +  3  +  9     (4) 
or  5  x  -  3  a;  =  3  +  9  (5) 

2a:=12 
a;  =  6 

Ans.    X  =  1. 
Ans.    X  =  6. 

6.  X  -\-  o  X  —  5  =  7. 

7.  5a; -4- 6  =  3a;. 

14.  The  parts  of  an  algebraic  expression  connected  by  the 
signs  +  or  —  are  called  terms.  Thus,  iu  Ex.  7,  there  are 
four  terms,  5  .x;  —  4,  —  6,  and  3  x. 

15.  The  equation  given  in  Ex.  1, 

5  X  -9  =  3  X  +  3  (1) 

it  will  be  seen  becomes  in  equation  (5) 

.5  ,r  -  3  a;  =  3  +  9  (5) 

That  is  the  terms  —  9  and  3  x  appear  in  equation  (5),  each 
on  tlie  side  opposite  to  that  on  whicli  it  appears  in  equa- 
tion (1),  and  with  an  opposite  sign.  For  convenience  this 
change  of  side  and  sign  is  called  transposition.  According  to 
this  any  term  can  be  erased  from  one  side,  provided  we  put 
it  on  the  other  side  with  the  opposite  sign. 


2. 

6x-^S  =  10x-l. 

3. 

8a;-15  =  6a;-3. 

4. 

9a;  +  8  =  10ar  +  6. 

5. 

7_4a;  =  7a;-4; 

10  ALGEBRA. 

8.    ^  +  4  =  16.  (1)        , 

If  we  subtract  4  from  both  sides  of  (1),  we  have, 

1=12  (2)  (Ax.  2) 

If  we  multiply  both  sides  of  (2)  by  2,  we  have, 

X  =  24  (Ax.  4) 

16.    This   last   process   is  called   ckariwj  the   equation  of 

fractions. 

9.^  +  5  =  1  +  7.  (1) 

If  we  multiply  (1)  by  9,  we  have, 

.3  X  +  45  =  X  +  6.3 

2a;=  18 

x  =  9 


1^- 1-^=1-^- 

13.   t?  +  6 

i 

11.   ^  +  8  =  15. 
o 

4a;       3a: 

7         7 

5 

=  ^%s 

(1) 

a- 
7 

=  8-5 

(2) 

X 

=  3 

(3) 

X  -- 

=  21 

Note.     It  is  often  better  as  iu  Ex.  13  above  to  unite  terms  before  clearing 
)f  fractions. 

•'"       -      ■'■       -  18.    ^'-lO^i^'-LS. 

o  10 

20.    1  +  1  +  1  =  26. 

4^2      5~  2      6  .■! 

22.    <,4+|-2^  =  6.  +  7i-^. 


4. 

5 

+    8: 

=  -  +  12. 

5. 

5x 
3 

2 

3' 

2.r         1 

=  :f  +  ,s 

l6. 

.3" 

-1  = 

6  +  2- 

REDUCTION   OF   SIMPLE   EQUATIONS.  11 


23.    ,-1  +  5  =  1  +  ^1  +  33  (1) 


Multiply  (1)  by  6. 

6x-^  +  2x^^, 

0                             1 

-  ox 

+  198 

(2) 

Transpose  and  unite. 

5         7 

(3) 

Multiply  (3)  by  7. 

21a-i|i--6x=1386 
5 

or                                  15  a:  -  1^  =  1386 

(i) 

Multiply  (4)  by  5. 

75  a;  -  42  a;  =  6930 

or                                             33  X  -  6930 

.r  =  210 

Note      We  might  have  multiplied  (1)  by 

6  X 

7X5, 

or  210,  at  once. 

24.  i-i  +  2ii='^-m- 

^ 

17.    N"iimber.s  are  often  grouped  by  different  forms  of  the 

Bracket,  (),[],{      },  f^i'd  the  Vinculum, .     Thus, 

{a  +  b  —  c),  I  a  +  b  —  c],  \a  +  b  —  c\,  a  +  h  —  c,  indicate 
that  a,  b,  and  —  c,  are  to  be  considered  as  one  whole,  and 
subjected  to  the  same  operation. 

The  expression  1(3  +  (8  —  3)  means  that  8  —  3,  or  5,  is  to 
be  added  to  16,  or  16  +  5  =  21.     The  result  is  the  same  if 
we  first  add  8  to  16  and  from  the  sum  subtract  3.     16  and  8 
are  24;  24  minus  3  are  21. 
That  is,  16  +  (8  -  3)  =  16  +  8  -  3  =  21 

The  expression  16  —  (8  +  3)  means  that  the  sum  of  8  and 
3  is  to  be  subtracted  from  16,  or  16  —  11  =  5.     In  this  case 
we  can  subtract  8  from  16,  which  leaves  8,  and  then  from  8 
subtract  3  more,  which,  as  before,  leaves  5. 
That  is,  16  -  (8  +  3)  =  16  -8-3  =  5 


12  ALGEBRA. 

The  expression  16  —  (8  —  3)  means  that  8  —  3,  or  5,  is  to 
be  subtracted  from  16,  or  16  —  5  =  11.  If  now  we  first  sub- 
tract 8  from  16  the  remainder  is  8  ;  but  we  are  not  to  sub- 
tract 8  but  3  less  than  8.  We  have  then  subtracted  3  too 
many,  and  therefore  the  remainder  8  must  be  3  too  small, 
and  the  true  remainder  must  be  8  +  3  =  11. 

That  is,  16  -  (8  -  3)  =  16  -  8  +  3  =  11 

The  principle  is  evidently  the  same  whatever  the  par- 
ticular numbers  may  be.  It  appears  then  that  if  a  paren- 
thesis is  removed  with  the  sign  +  before  it,  the  signs  of  all 
the  terms  remain  unchanged ;  but  if  the  sign  —  is  before  the 
parenthesis,  the  signs  of  each  term  within  the  parenthesis  is 
changed  from  +  to  — ,  or  —  to  +. 

Note.  The  first  term  in  a  parenthesis  having  no  sign  before  it  is  of 
course  +. 

18.  When  an  expression  of  several  terms  is  to  be  multi- 
plied or  divided,  each  term  must  be  multiplied  or  divided. 
Thus,  5  times  3  -f  7  is  15  -f-  35  =  50.  For  3  -h  7  =  10,  and 
5  times  10  is  50.  35  —  14  divided  by  7  is  5  —  2,  or  3.  For 
35  -  14  =  21,  and  21  divided  by  7  is  3. 

1.  4  (5  +  8)  =  20  +  32  =  52  =  4  •  13. 

2.  9  (8  -  3)  =  72  -  27  =  45  =  9  •  5. 

3.  (18  4-  12)  -^  6  =  3  -h  2  =  5  rrr  30  -f-  6. 

4.  (42  -  18)  ^  3  =  14  -  6  =  8  rr:  24  -^  3. 

The  line  between  the  numerator  and  denominator  of  a 

fraction  acts  also  as  a  vinculum. 

16  —  4 
Thus,         27  -    "  =  27  -  (4  -  1)  =  27  -  4  -I-  1  =  24 

4  ^ 

5.  38  +  ^~~~  =  38  +  (6  -  2)  =  42 . 

o 

6  o 


REDUCTION  OF  SIMPLE   EQUATIONS.  13 


19.    Find  the  value  of  x  in  the  following  equations 

X  +  5      X 
1-    ^-  +  6  =  '" 
Multiplying  (1)  by  6, 


l.-+^  +  t=2o--+^.  (1) 


•2x  +  10  +  -*^  -  =  150  -'3x-7i 
5 


Transposing  and  uniting, 

5x  +  ^'  =  62 


25  x  +  6  2  =  31  2'  =  310 

X  =  10 

9 

4  =  (1  +  7)  -  5.                              4. 

^-(M)-' 

3. 
5. 

2;r+l_            a.  +  3 
5       -" '       3     ■ 
9^-29      ^^      Gx  +  11 
4         --'             5        • 

6. 

117  -X      X  -  95 
^''-        4                 3       • 

7. 

5             ■               3         ■ 

8. 

-^-^r---''^r'-(- 

-!)• 

9. 

5  —  .r       4  a:  +  5       19  +  5  a; 
2                3        ==         4 

10 

7            9            2a;+  5      4a;-l 

6       5  .r  +  1  6  12 

Note.     Multiply  by  12;  then  transpose  and  unite. 

X  -\-  H\        »!  +  4  _  ^.         X  —  1 

"•    ^6  6~  -^^~      2     • 

Uniting  the  fractions  in  the  first  member, 

1^  =  2  =  21-^  +  1 
6  2       2 

x_39 

2~  ¥ 

x  =  39 


14 


[ 

ALGEBRA. 

12. 

x-6    ,    .        2:r-3             .. 

4--+'              3        -       ^^• 

13. 

22  -X      x-1       ^       S  +  x 

o       +      6      -'          o- 

•^2  -  r                                .  ,         3  +  x 
Note.      Transpose   — - —  aud  unite  it  with — . 

284  —  4  x      3  x-7o      2  a;  —  22 

14.  19 ^-  =  ^r i^- 

Note.     First  write  the  equation  thus  : 

284 -4x      .7-25      .r  -  11 


19 


3  2 


7  X  +  5      G  X  —  30 

15.  a; ,^ —  =  -r. ^  —  1  • 

Note.     First  multiply  by  7  ;  then  transpose  and  unite. 

20  a;  +  21  a:  —  12        ^ 

16.  Ax j'l—  +  1  =  ^^ 5. 

Note.     First  write  the  equation  thus : 

rhen  -x-^-4 

3  =  .r 
_     8a;-13      13 -3 a;       ,.  ^       ?>x+2 

17.      = ^TT, =r  h  a-  —   < ;; . 

5  10  o 

x  +  1       4a;  +  5_  5x— ^ 

^^'    ^6  5~-^~       4       ■ 

1Q     7       3^  +  ^"^       3(a;  +  5)_  5a^+J 
2        ~         4         ~  2 

21.    2_^«  +  23_-.  ^  ,  ^  2_  11  _  3^J 

^^    x  +  S      5x-6      4:X-6_  22-j^ 

6  <  o  3 


WRITTEN   EXERCISES.  15 


CHAPTEK    III. 

20.     ORAL    EXERCISES. 

1.  How  old  sliall  I  bt'  ill  a  yuur.s  if  I  ;uii  U5  years  old  now  ? 

2.  How  old  w;is  I  (i  yeurs  ago  if  1  iim  x  years  old  now  ? 

8.    How  old  shall  1  be  in  x  years  if  I  am  y  years  old  now  ? 

4.  If  John  has  a  cents  and  20  cents,  liow  many  cents  has  he  ? 

5.  If  1  am  -  11  years  old,  and  in  4  years  shall  be  twice  as  old 
^as  my  son,  how  old  is  my  son? 

6.  What  number  exceeds  a  by  4  '■. 

7.  Find  a  number  less  than  x  by  5. 

8.  If  a  is  a  number,  how  is  the  number  that  is  one  half  of  a 
to  be  expressed  ?     Three  times  as  great '.' 

9.  AVliat  number  exceeds  it  by  ??«.  ? 

10.  \i  X  is  one  part  of  12,  what  is  the  other  part  ? 

11.  If  of  two  factors  whose  product  is  24,  x  is  one,  what  is 
the  other  ? 

12.  If   <]r   is   the  quotient,   and   8   is  the  divisor,  what  is  the 
dividend  ? 

13.  If  (f  is  the  divisor,  y  is  the  quotient,  and  r  the  remainder, 
what  is  the  dividend  ? 

14.  AVliat  number  is  half  as  great  again  as  y  ? 

15.  If  35  contains  x  iive  times,  what  is  the  value  of  cc  ? 

16.  If  y  apples  cost  25  cents,  how  many  apples  can  be  bought 

for  II  ? 

17.  If  d  is  divided  into  two  parts,  and  8  is  one  of  them,  what 
is  the  other  ".' 

18.  If  6  X  is  the  product   of  two  numbers,  and  6  is  one  of 
them,  what  is  the  other  ? 


16  ALGEBRA. 

19.  If  I  walk  X  miles  in  5  hours,  how  many  miles  do  I  walk 
in  one  hour  ? 

20.  If  I  walk  X  miles  in  3  hours,  how  long  does  it  take  me 
to  walk  Iniile  ? 

21.  If  I  can  riiw  8  miles  in  h  hours,  what  is  my  rate  ?  How 
many  hours  does  it  take  me  to  row  one  mile  '/  How  many  miles 
can  I  row  in  one  hour  ? 

22.  If  I  can  row  mi  miles  in  //  hours,  what  is  my  rate  ? 

23.  If  X  men  can  build  a  wall  in  0  days,  how  long  will  it 
take  one  man  ? 

24.  If  9  men  can  dig  a  ditch  in  ./•  days,  how  long  will  it  take 
one  man  ? 

25.  If  X  men  can  dig  a  ditch  in  y  days,  how  long  will  it  take 
one  man  ? 

26.  If  Mr.  Smith  is  a  +  6  years  old  to-day,  how  old  was  he 
a  years  ago  ?     How  old  h  years  ago  ? 

27.  If  a  man  was  //  years  old  ten  years  ago,  how  old  will  he 
be  z  years  hence  ? 

21.     WRITTEN    EXERCISES. 

Should  any  pupil  find  difficulty  iu  answering  the  ques- 
tions, let  him  substitute  figures  for  the  letters. 

1.  A  boy  had  ./■  cents,  and  then  s])ent  5,  and  afterwards 
doubled  his  money.      How  many  rents  did  he  then  have  ? 

2.  A  faiinei'  having  2  ./■  sheej)  buys  24  more,  and  then  sells 
half  of  them.      How  many  sheep  has  he  left  '.' 

3.  John  is  four  years  older  than  Henry,  and  Henry  five  years 
older  than  James.  \i  x  years  is  Henry's  age,  what  represents 
the  ages  of  .lohn  and  James,  rcspecti-vely. 

4.  A  has  X  dollars,  and  P>  has  ¥liy  more  than  twice  as  much. 
r>  gives  A  -feO.  and  tbfii  each  doubles  the  money  he  has. 
How  nnicli  lias  each  '.'      How  um.-h  both  ? 

T).  Two  boys,  Robert  and  George,  had  together  *20.  At  the 
end  of  a  year  Robert  has  gained  *4   plus  what  he  had  at  first, 


WRITTEN   EXERCISES.  17 

and  George  has  gained  $25  less  than  three  times  what  he  had  at 
first.  If  X  represent  the  number  of  dollars  Robert  had  at  first, 
wliat  will  represent  the  number  each  has  at  the  end  of  the  year  ? 

6.  A  has  4  x  dollars,  and  B  half  as  much ;  then  A  lends  B 
IIO ;  then  both  A  and  B  double  what  they  have,  and  B  repays  A. 
Find  what  each  of  them  had. 

7.  If  a  merchant  begins  trade  with  x  dollars,  and  doubles  his 
stock  every  year,  lacking  1500,  what  will  he  ha\e  at  the  end  of 
the  third  year  ? 

8.  Monday  A  has  //  dollars.  Every  night,  including  Monday 
night,  he  gives  his  wife  ^2,  and  each  day  for  the  next  4  days 
doubles  what  he  has  left.  How  much  will  he  have  Saturday 
morning  ? 

9.  If  11  is  any  integral  number,  2  w  is  always  an  even  number. 
What  is  the  next  even  number  after  2  n  ''     The  next  before  it  ? 

10.  What  is  the  next  odd  number  after  2  7i  ?  The  second  odd 
number  ?     The  second  odd  number  before  2  n? 

11.  Find  the  sum  of  three  consecutive  odd  numbers  of  which 
the  middle  one  is  2  ?i  —  1. 

12.  A  is  X  years  old,  C's  age  is  three  times  B's,  and  B's  is 
twice  A's.     Find  C's  age. 

13.  If  a  man  was  m  years  old  x  years  ago,  how  old  will  he 
be  71  years  hence  ? 

14.  A  boy  is  n  years  old,  and  eight  years  hence  he  will  be 
half  the  age  of  his  father      How  old  is  the  father  now  ? 

15.  How  old  is  Mr.  Child,  if  y  years  ago  he  was  t  times  as 
old  as  his  son,  who  was  then  x  years  old  ? 

16.  What  number  exceeds  the  sum  of  a  and  b  by  their  differ- 
ence, if  a  is  greater  than  h  ? 

17.  If  a  horse  eats  o  bushels  of  oats  and  r  bushels  of  corn  a 
month,  how  many  bushels  of  oats  and  corn  will  he  eat  in 
m  months  ? 

18.  If  a  train  runs  ///  miles  in  Ji  hours,  what  is  its  rate  ? 
Express  it  in  two  ways.     (See  Ex.  21,  §  20.) 


18  ALGEBRA. 

19.  If  a  post  8  X  feet  in  length  is  half  in  the  water,  and  one 
eighth  in  the  mud,  and  the  rest  above  the  water,  how  many  feet 
are  above  the  water  ? 

20.  How  far  will  a  person  walk  in  50  minutes,  if  he  walks 
m  miles  in  h  hours  ? 

21.  How  long  will  it  take  a  person  to  walk  a  miles,  if  he 
walks  25  miles  in  h  hours  ? 

22.  How  long  will  it  take  x  men  to  cut  y  acres  of  grass,  if 
each  man  cuts  z  acres  a  day  ? 

23.  A  and  B  start  from  the  same  point  and  walk  in  exactly 
opposite  directions  at  the  rate  of  a  and  h  miles  an  hour,  respec- 
tively. How  far  apart  will  they  be  at  the  end  of  one  hour  ? 
5  hours  ?     15  minutes  ? 

24.  If  A  and  B  start  from  the  same  point  and  walk  in  the 
same  direction  at  rates,  resjDectively,  of  a  and  h  miles  an  hour, 
how  far  apart  will  they  be  at  the  end  of  one  hour  ?  At  the  end 
of  2  hours  and  20  minutes  ? 

25  If  A  can  do  a  piece  of  work  in  d  days,  what  part  of  the 
work-  can  he  do  in  one  day  ?     In  two  days  '.'     In  h  daj^s  ? 

26.  If  a  man  can  do  a  ])iece  of  work  in  -  days,  what  part  of 
the  work  can  he  do  in  one  day  ? 

27.  A  can  do  a  piece  of  work  in  (i  days,  and  B  in  i  days. 
What  part  of  the  work  can  they  do  together  in  one  day  ?  How 
long  will  it  take  them  to  do  the  whole  work  '.' 

28.  If  a  pipe  discharges  x  gallons  of  water  in  //  hours,  how 
many  does  it  discharge  in  one  hour  ?  How  long  does  it  take  to 
discharge  one  gallon  ? 

20.  A  can  do  a  piece  of  work  in  a  days,  B  in  h  days,  and 
C  in  c  days.  What  part  of  the  work  can  they  do  together  in 
one  day  ?  How  long  will  it  take  them,  working  together,  to  do 
the  work-  ? 


PKOBLEMS.  19 

PROBLEMS 

PRODUCING   SIMPLE   EQUATION!*   CONTAINING   BUT   ONE   UNKNOWN   NUMBER. 

22.    The  Solutiuii  ut'  a  Problem  in  Algebra  consists : 

1st.    In  reducing  the  statement  to  the  form  of  an  equation  ; 

2d.  In  reducing  the  equation  so  as  to  tind  the  value  of 
the  unknown  numbers. 

1.  A  father  and  .son  liave  pro])erty  worth  166,  and  tlic 
father's  part  is  double  the  son's.      What  is  the  part  of  eacli  ? 

Ans.     Son's  part,  $22.     Father's  part,  $44. 

2.  The  sum  of  three  numbers  is  49.  The  greatest  is  four 
times  the  least,  and  the  intermediate  number  is  twice  the  least. 
What  are  the  numbers  ?  Ans.     7,   14,   28. 

3.  Two  boys  have  $24.  If  the  first  has  double  the  second, 
how  many  dollars  has  each  ? 

Ans.      The  first,  $8;  the  second,  $16. 

4.  A  father's  age  is  three  times  that  of  his  son,  and  their 
ages  added  together  amount  to  48  years.      How  old  is    the  son  ? 

5.  A  carriage-horse  cost  three  times  as  much  as  the  carriage, 
and  the  price  of  the  two  was  $360.     Find  what  each  cost. 

6.  In  a  school  of  84  pupils,  twice  as  many  study  geography 
as  grammar,  and  twice  as  many  study  arithmetic  as  geography. 
How  many  pupils  are  there  in  the  respective  classes  ? 

Ans.     12,  24,   and  48. 

7.  The  distance  between  two  jdaces.  A  and  B,  is  four  times 
the  distance  between  B  and  C.  If  the  difference  of  these  dis- 
tances is  51  miles,  find  the  number  of  miles  from  A  to  B. 

8.  From  two  towns,  77  miles  apart,  two  carriers  at  the  same 
time  start  toward  each  other  at  the  rate  of  o  and  6  miles  an 
hour,  respectively.      In  how  many  hours  will  they  meet  ? 


20  ALGEBRA. 

0.  At  tho  end  of  the  fourth  day  the  captain  of  a  vessel,  sail- 
ing from  Boston  to  London,  found  that  he  had  sailed  240  miles. 
The  second  day  he  sailed  twice  as  far  as  the  first,  and  the  third 
as  far  as  the  first  and  second  days  together,  but  the  fourth,  by  a 
severe  storm,  he  was  driven  back  as  many  miles  as  he  sailed  the 
second  day.      How  many  miles  did  he  sail  the  first  day  ? 

Ans.     60. 

10.  A  drover  sold  7  sheep  and  5  lambs  for  $57,  receiving 
twice  as  much  for  a  sheep  as  for  a  lamb.  How  much  did  he 
receive  for  each  ? 

11.  Three  men  hire  a  pasture  for  the  summer  for  $81.  The 
first  puts  in  two  cows;  the  second,  three;  and  the  third,  four.  If 
the  cows  remain  in  the  pasture  the  same  length  of  time,  how 
much  ought  each  to  pay  ? 

12.  Three  men,  A,  B,  and  C,  built  57  rods  of  fence.  A 
built  5  rods  a  day;  B,  3;  and  C,  2.  C  worked  twice  as  many 
days  as  B,  and  B  twice  as  many  as  A.  How  many  days  did 
each  work  ?  Ans.     A,  3  days;    B,  6  days;    C,  12  days. 

13.  The  difference  between  two  numbers  is  7,  and  their  sum 
is  23.     What  are  the  numbers.  Ans.     8  and  15. 

14.  To  four  times  a  certain  number  I  add  12,  and  obtain  32. 
What  is  the  number  ? 

15.  Two  persons,  A  and  B,  divide  .$50  between  them,  so  that 
A  has  ^6  more  than  B.     What  are  their  shares  ? 

16.  Three  checks  are  together  worth  $80.  The  first  is  worth 
$25  less  than  the  second,  and  $20  more  than  the  third.  What 
are  their  values  ?  Ans.     $25,   $50,   $5. 

17.  A  father  left  his  three  sons  $10000.  the  eldest  to  have 
$2000  more  than  the  second,  and  $3000  more  than  the  third. 
Find  the  share  of  each. 

18.  In  a  hamlet  containing  90  persons,  there  are  4  less  men 
than  women,  and  10  more  children  than  adults.  How  many 
men,  women,  and  children  are  there  ? 


PROBLEMS.  21 

19.  A  has  two  dollars  less  than  three  times  what  B  has,  and 
they  both  together  have  $62.     What  has  each  ? 

20.  Divide  $300  among  A.  B,  and  C,  so  that  A  shall  have 
twice  as  much  as  B,  and  B,  -^20  more  than  C. 

Ads.      a,  S160;    B,  $80;    C,  $60. 

21.  A  man  walks  S  miles,  then  goes  a  certain  distance  bj' 
coach,  and  then  three  times  as  far  by  train  as  by  coach.  If  the 
whole  journey  is  88  miles,  how  far  did  he  travel  by  train  ? 

22.  Find  a  number  such  that,  if  10  is  taken  from  its  double, 
and  20  from  the  double  of  the  remainder,  there  will  be  40  left. 

Ans.     20. 

23.  A  father  said  to  his  son,  "3  j^ears  ago  I  was  three  times 
as  old  as  you,  but  in  9  years  I  shall  be  exactly  twice  as  old." 
What  were  their  ages  ? 

Let  .r  =  no.  of  years  in  the  son's  age; 

then  (x  -  3)  3  +  3  =  "  "  father's  age. 

X  +  9  —  "  "  son's  age  9  years  hence. 

(a;  —  3)  3  -I-  12  =  "  "  father's  age  9  years  hence. 

(x-  3)  3  +  12  =  2  (r  +  9) 
3  X  +  3  =  2  .r  +  18 

X  =  15  no.  of  years  in  the  son's  age. 

(x  -  3)  3  +  3  =  39  "  "  father's  age. 

24.  A  certain  number  of  two  figures  whose  sum  is  8  will  have 
the  order  of  the  figures  reversed  if  18  is  added  to  it.  "What  is 
the  number  ?  Ans.     35. 

Let  X  =  the  tens'  figure ; 

then  8  —  X  =z  the  units'  figure. 

l(»x  +  8  —  X  =  9 X  +  8  =  the  number; 
and  when  the  order  of  the  figures  is  reversed, 

10  (8  -  x)  +  X  =  80  -  9  X  =  the  number. 
9  j;  +  8  -f-  18  =  80  -  9  X 
18x  =  54 

X  =  3  the  tens'  figure. 
8  —  X  =  5  the  units'  figure. 


22  ALGEBRA. 

25.  A  train  carries  185  travellers  from  Liverpool  to  London. 
The  number  of  travellers  of  the  third  class  is  19  more  than  the 
number  of  the  other  two  together,  and  of  the  second  class, 
13  more  than  of  the  first  class.      Find  the  number  of  each  class. 

Ans.     35;   48;    102. 

26.  A  pupil  who  said  he  had  received  the  first  prize  in  Arith'- 
metic  was  asked  how  many  other  prizes  he  had  received.  He 
replied:  "If  from  4  times  the  number  of  my  other  prizes,  you 
subtract  3,  then  add  5  more  than  double  this  number,  the 
result  will  be  14."     Find  the  number  of  all  his  prizes.     Ans.  3. 

27.  "What  is  the  date  of  3-our  birth  ?"  said  a  pupil  to  his 
master.  The  master  answered:  "The  sum  of  the  four  figures 
is  18;  the  tens'  figure  is  2  less  than  the  hundreds'  and  3  more 
than  the  units'."     What  was  the  date  ?  Ans.     1863. 

Note.     The  thousands'  figure  is  of  course  known. 

28.  A  person  gave  $8  to  4  families.  To  the  second,  $0.80 
less  than  twice  what  he  gave  to  the  first;  to  the  third,  $1.40 
less  than  three  times  Avhat  he  gave  to  the  first;  and  to  the 
fourth,  10.60  more  than  twice  what  he  gave  to  the  first.  How 
much  did  he  give  to  each  ?  Ans.      $1.20;  $L60;  $2.20;   $3. 

29.  A  student  bought  four  books.  The  price  of  the  second 
was  $0.60  less  than  twice  the  price  of  the  first;  of  the  third, 
$0.20  more  than  twice  the  price  of  the  first ;  of  the  fourth,  $0.80 
less  than  three  times  the  price  of  the  first;  and  the  sum  paid 
for  the  first  three  was  $2.80  more  than  the  price  of  the  fourth. 
Find  the  price  of  each  book.     Ans.   $1.20;  $1.80;  $2.60;  $2.80. 

30.  A  boy  started  from  home  to  walk  to  Boston.  The  sec- 
ond day  he  walked  10  miles  less  than  twice  the  distance  he 
walked  the  first  day ;  the  third  day,  three  halves  as  much  as  the 
first  day;  the  fourth  day,  16  miles  less  than  three  times  the 
distance  he  walked  the  first  day,  and  reaches  the  city.  More- 
over the  distance  walked  on  the  second  and  third  days  together 
was  a  mile  less  than  on  the  first  and  last  days  together.  How 
far  was  his  home  from  Boston  ?  Ans.     79  miles. 


PROBLEMS.  23 

31.  Divide  5};760  among  A,  B.  and  C,  .so  that  B  may  have 
150  more  than  A  and  $135  less  than  C. 

32.  A  man  paid  $22.20  to  two  boys  who  had  worked  tlie 
first,  12  days,  and  the  second,  15.  The  second  received  $0.40  a 
day  more  than  the  first.      Find  the  daily  wages  of  each. 

Ans.     $0.60;    $1. 

33.  A  man  paid  $40  with  07  pieces  of  silver,  dollars  and 
quarters.      How  many  of  each  did  he  use  ? 

Ans.     31  dollars,  and  36  quarters. 

34.  Two  persons  have  the  same  annual  salary.  The  first 
saves  1  of  his,  and  the  second  spends  $160  more  than  the  first 
animally,  and  at  the  end  of  3  years  is  in  debt  $170.40.  AVhat 
is  this  salary  ?  Ans.     $516. 

35.  A  sum  of  mone}'  is  given  to  A  and  B  in  such  a  way  that 
A  has  I  as  much  as  B,  and  y'o  ^^  ^'^  money  plus  y^^f  of  B's  is 
$21.     Find  the  share  of  each  ? 

36.  Two  casks  contain  one,  90  gallons,  and  the  other,  110 
gallons  of  oil.  The  same  quantity  is  drawn  from  each,  and 
then  the  first  contains  ^  as  much  as  the  second.  What  is  the 
quantity  drawn  out  ? 

37.  A  journeyman  agreed  to  work  during  the  month  of  Sep- 
tember (four  Sundays  excepted)  for  $3.50  a  day;  but  for  every 
day  he  was  idle  he  was  to  forfeit  $1.50.  He  received  $61.  How 
many  days  was  he  idle  ?  Ans.      6. 

38.  How  many  days  could  the  journeyman  named  in  Ex- 
ample 37  have  w^orked  and  yet  have  been  entitled  to  no  pay  ? 

Ans.    181. 

39.  Two  towns,  A  and  B,  are  100  miles  apart.  At  A,  coal 
costs  $5  a  ton,  and  at  B,  $5.25  a  ton.  If  the  freight  is  a  cent 
and  a  quarter  a  ton  a  mile,  at  what  point  between  A  and  B 
does  coal  cost  the  same  ?  How  does  this  cost  compare  with  the 
cost  at  other  points  between  A  and  B  ? 

Ans.      1st.    60  miles  from  A  :     "Jd.    Dearest. 


24  ALGEBRA. 

40.  In  a  certain  factory  $4560  are  paid  each  week  as  wages. 
The  workmen  are  divided  into  three  classes.  The  first  class 
receive  $6  each  a  week;  the  second,  $7;  and  the  third,  |>8. 
There  are  4  of  the  first  class  for  12  of  the  second,  and  4  of  the 
second  for  o  of  the  third.  Find  the  number  of  workmen  in  each 
class.  Ans.     1st  class,  80;  2d,  240;  3d,  300. 

41.  A  father  43  years  old  has  two  sons,  10  years  and  6  years 
old,  respectively.  How  long  ago  was  the  father  4  times  as  old 
as  the  sum  of  the  ages  of  his  two  sons  ? 

42.  A  man  engaged  a  coachman  for  |?300  a  year  and  his 
livery.  At  the  end  of  8  months  the  coachman  left,  receiving 
$188  and  keeping  the  livery.     What  was  tlie  cost  of  the  livery  ? 

Ans.     136. 

43.  There  are  two  pieces  of  land  whose  combined  areas  are 
33|  acres,  and  |  of  one  is  equal  to  {^  of  the  other.  Find  the 
area  of  each  ?  Ans.     18  acres,  and  15f  acres. 

44.  Two  casks  contain  together  55  gallons  of  kerosene.  If 
a  third  part  of  the  first  and  a  fifth  part  of  the  second  are  drawn 
out,  the  casks  will  contain  an  equal  amount.  How  many  gal- 
lons are  there  in  each  ? 

45.  Two  boys  save,  one  a  third,  and  tlie  other  a  fourth  of 
his  income.  The  sum  of  their  gain  is  $80  and  the  sum  of  their 
income  is  $270.     What  is  the  income  of  each  ? 

Ans.     $150;  $120. 

46.  Two  persons,  A  and  B,  receive  by  will,  together,  $3660. 
A  spends  f  of  his,  and  B  f  of  his,  and  then  A  has  twice  as 
much  as  B.     How  much  did  each  receive  ? 

Ans.     A,  $2400;  B,  $1260. 

47.  A  dealer  in  grain  wished  to  fill  a  bin  that  holds  217 
bushels  with  a  mixture  of  oats  and  barley.  The  oats  cost  $0.45, 
and  the  barley  $0.52,  a  bushel.  How  many  bushels  of  each 
must  he  take  in  order  that  a  bushel  of  the  mixture  may  cost 
10.50  a  bushel  ?  Ans.     Oats,  62;  barley,  155. 


PROBLEMS.  25 

48.  Divide  $1298  among  4  persons  so  that  the  first  shall 
have  $20  more  than  tlie  second;  the  second,  $48  more  than  the 
third;  and  the  third,  $70  more  than  the  fourth. 

Ans.     1st,  $381;    2d,  $361;    3d,  $313;    4th,  $243. 

49.  Divide  $2000  among  4  persons  so  that  tlie  first  may 
have  $400  less  than  twice  what  tlie  second  has;  the  second, 
$600  less  than  three  times  as  much  as  the  third  has;  and  the 
tliird,  $800  less  than  six  times  as  much  as  the  fourth  has. 

Ans.     1st,  $800;    2d,  $600;    3d,  $400;    4th,  $200. 

50.  Divide  $360  among  3  persons  so  that  the  second  may 
have  $30  more  than  f  as  much  as  the  first,  and  the  third,  $24 
less  than  f  as  much  as  the  second. 

Ans.     1st,  $195;  2d,  $108;  3d,  $57. 

51.  Divide  $1120  among  5  persons  so  that  the  second  may 
have  $40  more  than  double  the  first;  the  third,  $80  less  than 
three  times  the  first;  the  fourth,  $30  more  than  half  as  much  as 
the  second  and  third  together;  and  the  fifth,  $95  more  than  a 
fourth  as  much  as  the  four  others  together. 

Ans.    1st,  $100;  2d,  $240;  3d,  $220;  4tli,  $260;  5th,  $300. 

52.  A  sum  of  money  was  divided  among  4  persons  so  that 
the  first  had  ^  of  the  whole;  the  second,  |  of  the  remainder; 
the  third,  |  of  the  second  remainder;  and  the  fourth,  the  rest, 
or  $480.     What  was  the  sum?  Ans.      $1800. 

53.  Divide  $9  among  9  persons :  1  man,  3  women,  and  5 
children,  in  such  a  way  that  the  man  may  have  J  as  much  as  a 
woman,  and  a  woman  f  as  much  as  a  child. 

Ans.      The  man,  $2.25;   a  woman,  $1.35;  a  boy,  $0.54. 

54.  From  Fitchburg  to  Boston  by  rail  is  50  miles.  If  coal 
can  be  bought  in  Boston  at  $6.60  a  ton,  and  in  Fitchburg  at 
16.75,  and  it  costs  for  freight  a  cent  and  a  half  a  mile  a  ton, 
at  what  point  on  the  line  would  coal  cost  exactly  the  same 
whether  it  came  from  Boston  or  Fitchburg  ? 

Ans.     30  miles  from  Fitchburg. 


26  ALGEBRA. 

55.  A  box  contains  115  in  silver,  dollars  and  quarters.  If 
there  are  21  pieces  of  money,  how  many  of  each  kind  are  there  ? 

Ans.      13  dollars;   8  quarters. 

56.  The  toll  on  a  certain  bridge  for  a  man  on  foot  is  1  cent; 
for  a  man  on  horseback,  2  cents;  for  a  carriage  with  one  horse, 
3  cents ;  and  for  a  carriage  with  two  horses,  4  cents.  On  a  cer- 
tain day  there  passed  over  the  bridge  carriages  with  two  horses, 
f  as  many  as  carriages  with  one  horse ;  carriages  with  one  horse, 
j\  as  many  as  men  on  horseback;  men  on  horseback,  /^  as  manj'^ 
as  men  on  foot;  and  the  receipts  were  $57.12.  How  many  of 
each  variety  were  there  ? 

Ans.  3564  men;  660  men  on  horseback;  180  carriages  with 
one  horse;   72  carriages  with  two  liorses. 

57.  A  merchant  increased  liis  capital  each  year  by  a  third  of 
what  he  had  at  the  beginning  of  the  year,  and  at  the  end  of  each 
year  took  out  for  his  expenses  $1000.  At  the  beginning  of  the 
fourth  year  his  capital  is  double  the  original  capital.  What 
was  the  original  capital  ?  Ans.     -f  11100. 

58.  A  banker  at  the  end  of  a  year  had  increased  his  capital 
by  f  of  itself;  during  the  second  year  he  lost  ^  of  what  he  had 
at  the  beginning  of  the  year  ;  during  the  third  year  he  added 
an  amount  equal  to  yV  of  his  original  capital  ;  and  during  the 
fourth  year  the  gain  was  equal  to  the  gain  of  the  first  three 
years  together.  He  then  had  $28560.  What  was  his  original 
capital?     .  Ans.   $20160. 

59.  A  father  and  his  two  sons  were  paid  for  building  a  wall 
in  such  a  way  that  the  father  received  as  much  an  hour  as  both 
sons,  and  the  elder  son  5  cents  an  hour  more  than  the  younger. 
The  father  worked  6  hours  and  30  minutes;  the  elder  son,  5 
hours  and  30  minutes;  and  the  younger,  4  hours  and  30  minutes. 
The  sum  received  was  $2.90.  What  were  the  hourly  wages  of 
each?       Ans.    Younger  son,  $0.10  ;  elder,  $0.15;  father,  $0.25. 

60.  A  father  is  35  years  old  and  his  son  6.  How  long  be- 
fore the  father  will  be  three  times  the  age  of  his  son  ? 


PROBLEMS.  27- 

61.  Three  partners  divide  the  profits  of  the  year  so  that  the 
first  has  $20  more  than  half  the  profits ;  the  second  has  a  fifth 
of  the  remainder  and  .f80;  and  the  third,  the  rest,  or  fl04. 
What  was  the  amount  of  the  profits  ?  Ans.     $500. 

62.  A  florist  started  for  market  with  roses  which  he  expected 
to  sell  at  5  cents  apiece.  On  the  way  he  lost  8  roses,  hut  found 
that  hy  selling  the  rest  at  6  cents  apiece  he  should  have  the 
same  amount  of  money  as  expected  when  he  started.  How 
many  roses  did  he  have  when  he  started  ? 

63.  A  boy  when  asked  his  age  said,  "In  16  years  I  shall  be 
three  times  as  old  as  I  was  2  years  ago."     How  old  was  he  ? 

64.  A  person  spent  each  day  half  the  money  he  had  left  the 
day  before,  plus  half  a  dollar,  and  at  the  end  of  the  third  day 
had  nothing  left.      How  much  money  did  he  have  at  first  ? 

65.  A  market-man  sells  to  one  person  half  of  his  eggs  plus  15 
eggs ;  to  a  second,  ^  of  the  remainder  plus  10  eggs ;  to  a  third, 
^  of  the  rest  plus  9,  and  has  none  left.  How  many  did  he  have 
at  first  ? 

66.  "How  old  are  you  ?"  asked  a  son  of  his  father.  "In  a 
year,"  answered  the  father,  "I  shall  be  three  times  as  old  as 
you  will  be,  and  in  19  years  twice  as  old  as  you  will  be."  How 
old  were  the  father  and  the  son  ? 

Ans.     Son,  17  j^ears  ;  father,  53. 

67.  A  man  sold  a  horse  for  25%  more  than  he  paid  for  it. 
What  he  paid  and  what  he  received  were  together  $675. 
What  did  he  pay  ? 

68.  A  man  invested  \  of  a  certain  sum  of  money  at  3|% ,  J  at 
4%,  and  the  rest  at  4J%.  The  annual  interest  of  the  three 
parts  amounted  to  $166.     What  was  the  sum  invested  ? 

69.  What  number  must  be  added  to  the  terms  of  f  that  it  (f ) 
may  become  equivalent  to  §§^  ? 

70.  What  one  number  must  be  added  to  the  two  terms  of  |f 
that  it  ma}'  become  equivalent  to  f  f  ? 


28  ALGEBRA. 

71.  Two  men  start,  the  one  at  6  A.  M.  and  the  other  at  8.30 
A.  M.,  and  walk  directly  toward  each  other.  The  first  goes  at 
the  rate  of  a  mile  in  24  minutes,  and  the  second,  a  mile  in  20 
minutes.  When  they  meet  they  have  travelled  equal  distances. 
What  is  the  distance  between  the  points  of  starting  ? 

72.  A  contractor  pays  daily  to  55  workmen,  men  and  hoys, 
$81.25;  to  each  man,  fl.75,  and  to  each  hoy,  fO.75.  How 
many  men  and  how  many  hoys  does  he  employ  ? 

73.  A  father  is  30  years  older  than  his  son,  and  in  4  years 
will  he  4  times  as  old.      Find  the  age  of  each. 

74.  $33456  are  put  at  interest  in  two  parts,  such  that  the 
interest  of  the  first  part  for  3  years  at  4%  is  double  the  inter- 
est of  the  second  part  for  7  years  at  5%.     Find  the  parts. 

75.  $12200  are  put  at  interest  in  two  sums,  one  at  4J%,  and 
the  other  at  3^%,  so  that  the  annual  interest  is  $489.  What 
are  the  two  parts  ?  Ans.    $6200  at  4^%  ;  $6000  at  3^%. 

76.  $9000  are  put  at  interest  in  two  parts,  one  at  5i%,  and 
the  other  at  4%,  so  as  to  give  an  annual  revenue  of  $400.50. 
What  are  the  two  parts  ?       Ans.    $2700  at  5h%  ;  16300  at  4%. 

77.  A  person  puts  |  of  a  certain  sum  of  money  at  interest  at 
4%,  and  the  rest  at  5%,  the  interest  for  72  days  is  $84.83. 
What  is  the  sum  ?     (Count  360  days  a  year.)         Ans.   $99880. 

78.  A  person  puts  |  of  a  certain  sum  of  money  at  5%,  and 
the  rest  at  4J%,  and  receives  at  the  end  of  a  year  for  interest 
and  principal  $3145.     What  was  the  principal  ?      Ans.    $3000, 

79.  A  certain  principal  at  8  %  gives  $232  more  interest  annu- 
ally than  another  principal  $1800  less,  at  6%.  What  are  the 
two  principals  ?  Ans.    At  8%,  $6200;  at  6%,  4400. 

80.  A  man  who  has  $24000  spends  a  jjart  in  the  purchase  of  a 
house,  the  rest  he  puts  at  interest,  one  third  at  4%,  and  two 
thirds  at  5%,  and  thus  has  an  income  of  $784  annually.  Find 
the  price  of  the  house,  and  the  two  sums  at  interest. 

Ans.    House,  $7200.      At  int.  4%,  $5600;  5%,  11200. 


SIMPLE  EQUATIONS.  29 


CHAPTER    IV. 

SIMPLE    EQUATIONS    CONTAINING    TWO    OR   MORE 
UNKNOWN    NUMBERS. 

ELIMINATION. 

23.  Elimination  is  the  nietliod  of  deriving  from  the  given 
equations  a  new  equation,  or  equations,  containing  one  (or 
more)  less  unknown  number.  The  unknown  number  thus 
excluded  is  said  to  be  eliminated. 

24    Equations  containing  only  two  Unknown  Numbers. 

1      Solve  P"  +  ^?^  =  '^l-  ^^^ 

y  =  ^^  (3)  3x  +  4(^L^)  =  -29  (4) 

15x+  124-8a;=  145  (5) 

y  =  ^l^  =  5  (7)  .  =  3  (fi) 

Transposing  2x  in  (1),  and  dividing  by  5,  we  have  (3),  which  gives 
an  expression  for  the  value  of  y.  Substituting  this  value  of  ?/  in  (2), 
we  have  (4),  which  contains  but  one  unknown  number  ;  that  is,  y  bas 
been  eliminated.  Reducing  (4)  we  obtain  (6),  or  x  =  3.  Substituting 
this  value  of  x  in  (3),  we  obtain  (7),  oi-  y  =  5.     Hence  the  following 

Rule. 

Find  an  expression  for  the  value  of  one  of  the  unknoimi 
numhers  in  one  of  the  equations,  and  substitute  this  value 
for  the  same  unknown  number  in  the  other  equation. 

This  method  of  elimination  is  called  substitution. 

Note  1  After  eliminating,  the  resulting  equation  is  reduced  as  shown 
in  Arts.  13-19.  The  value  of  the  unknown  number  thus  found  must  be  sub- 
stituted in  one  of  the  equations  containing  tlie  two  unknown  numbers,  and 
this  reduced  as  shown  in  Arts.  13-19. 


30  ALGEBRA. 

Solve  the  following  equations  b}^  substitution. 
2    f  5  .ic  -  4  //  =  -  2.  ,.    f  7  x  +  2^^  =  24. 

^'   ( 4  X  —  (>  y  =  -  10.  "  \qx  -    y  =  l. 

Qx-     y  =  l\.  ..    j  8a:  +    13  =  55/. 


5  a;  4-  3  2/  =  36.  '   ( 3  y  -  4  ic  =  11. 

jGcf-     y  =  0.  7    (3a:-     2/ =  20. 


<  3  .T  +  2  y  =  15.  (  5  a-  +  2  y  =  81. 

Note  2.  Tlie  pnpil  should  verify  his  results.  It  is  not  sufficient  to  substi- 
tute the  results  iu  one  equation.  Both  answers  may  he  wrong  and  yet  verify 
in  one  of  the  given  equations. 


(20^  +  52/ =  23. 
i  4  a;  +  3  ?/  =  25. 

11. 

\  7.T-43/  =  20. 

(3  a;  +  25/ =  42. 

1  5  a;  +  4  ^  =  53. 
l2a;  +  3y  =  31. 

12. 

(  13  .X  -  3  y  =  6. 

1    8  a;  +  5y  =  79. 

\4x-y     =3. 
(  3  .T  +  2  y  =-.  5. 

13. 

14y    +7.r  =  87. 

(3  a;    +2^  =  41. 

Solve  1^^  +  ^.'/  = 

=  35. 

(1) 

(3a:  +  2,y  = 

=  21. 

(2) 

]2x+  15i/=  105 

(3) 

12  x+    81/rz  84 

(5) 

3x  +  6  =  21 

7  7/ =  21 

0) 

V=    3 

(6) 

a;  =    5 

(8) 

10. 
14. 


It'  we  multiply  (1)  by  3  anrl  (2)  by  4,  we  have  (3)  and  (4),  in  which 
the  coefficients  of  x  are  equal  ;  subtracting  (4)  from  (3),  we  have  (5), 
which  contains  but  one  unknown  number.  Reducing  (5)  we  have  (6), 
or  y  =  3  ;  substituting  this  value  of  y  in  (2),  we  obtain  (7),  which 
reduced  gives  (8),  or  x  —  5.     Hence  the  following 

Rule. 

Multiplii  or  divide  the  eejuations  so  that  the  coefficients  of 
the  iinknown  imviher  to  be  eliviincded  shall  become  er^ual  ; 
then,  if  the  signs  of  this  number  are  alike  in  both,  subtract 
one  equation  from  tJie  other  ;  if  u.nlike,  add  the  tvv  equaticms 
together. 

Tliis  inetliod  of  eliniiiiatioii  is  called  comb iiud ion. 


ELIMINATION.  31 

Note.  The  least  multiplier  for  each  equation  will  be  that  which  will  make 
the  coethcient  of  the  unknown  number  to  be  eliminated  the  least  common 
multiple  of  the  two  coefficients  of  this  number  in  the  given  equations.  It  is 
always  best  to  eliminate  that  unknown  number  whose  coefficients  can  most 
easily  be  made  equal. 

Solve  the  following-  equatious  by  combination : 
-j^g    5    6x  —  T)!/  ^  10.  ^g     (  8x  —  3  1/  =  23. 

'    (  5x  +  4// =  32. 

i      ^  _    ^y  -  o 
16.  -]    ---r-y-—  19.    !      3         4~ 


17. 


()X 

— 

'■'y  = 

:   10. 

O  X 

+  Si/  = 

:  37. 

3x 

+ 

-Z/  = 

:   20. 

7  .'■ 

- 

••^y  = 

:  0. 

f  11a.- 

+ 

^i/  = 

:19. 

4:X 

+ 

3//  = 

:   10. 

20. 


X  '/  ^ 

o          6 

f  ^  +  f  y  =  ^■ 


25.    Solve  the  following  equations: 

Note.  Which  method  of  elimination  should  be  used  depends  upon  the 
relations  of  the  coefficients  to  each  other.  That  one  which  will  eliminate 
the  number  desired  with  the  least  work  is  the  better. 

(3a;  +  4^  =  34.  (      ^      ,      y  -rr 


[  6  2l'  +  3  ?/  =  33. 

f  4  ic  -  3  //  =  0. 
1  5x  +  2v^6d. 


{":  +  ■!. 


^  + 


o 


X  y 


o. 


o 


4   -    5  -  ^• 


I      3     '     7  I  10 

^   +    ^--  r^^  +  ^^-59 

hx      3>j      ^ 


4.+    V  =  n2.  ^^       ^ 


4 


!  -^^  +  a;  =  lo. 
+  a;=  15. 


l      5 

^      ^       y         Q  (X+1 ,     ^  ^ 

3+4-9.  ^^        ___  +  4^^5. 

3x-  +  4y=    109.  (4x-2y  =  31 


32  ALGEBRA. 


9 


Note  .     lu  the  first  equation  above  the  -~  on  the  left  is  balanced  by  the  — 
on  the  right,  and  the  equation  can  be  written  at  once 
a-      -r      „       „.  .r       11 


I- 

+  y          a;-2/       , 

12. 

5                  5      -^• 

-  //           a-  +  7/        13 
3         '        4      ~  4  ■ 

13. 

r 

J 

I 

x  —  ^!/.l/       ^'  +  LI 
4        ^5-2 
X         y        X       y 
11  ^10      4       4 

1 

2.-     •"-«.2,.- 

14. 

1 
I 

3.-2%+«  =  2.+ 

-27. 


26.    Simple  Equations  containing   more   than  two   Unknown 
Numbers. 

i     X  +     y  +     z  —  Q>. 
1.     -]  2  cc  +  3  7/  +  4  ^  =  20. 

x  +  yi-z  =  G{\)      2x  +  2ij  +  Az  =  20  (2)      .3.r -4  ?/ +    6  :,' =  13     (.3) 

2x  +  2y  +  2.::=12  (4)      3.r+3y+    3s  =18     (5) 

y  +  2z=    8  (6)  -1y+    32  =  -5  (7) 

7y+14s  =  56     (8) 

a;+2  +  3  =  6     (13)         i/  +  6=8     (11)  I7s  =  51     (9) 

a:=l      (14)  2/ =  2     (12)  z=    3    (10) 

Multiplying  equation  (1)  by  2  gives  eciuation  (4),  which  we  subtract 
from  (2).  and  obtain  (6)  ;  multiplying  (1)  by  3  gives  (5),  and  subtract- 


ELIMINATION.  33 

ing  (5)  from  (3)  gives  (7).  We  have  now  two  equations,  (6)  and  (7), 
containing  but  two  unknown  quantities.  Multiplying  (6)  by  7,  we 
obtain  (8),  and  adding  (7)  to  (8),  we  obtain  (9),  which  reduced  gives 
z  =  3.  Substituting  this  value  of  z  in  (6),  and  reducing,  we  obtain 
y  =  2.  Substituting  these  values  of  y  and  z  in  (1),  and  reducing,  we 
obtain  x  —  1. 


C2x-3y 
2.    Solve  j             3x 

2a;-3^  +  43  =  5       (1) 
v  +  4s  =  7 
2a:-4?/  =  -2  (4) 
6x  +  4y  =  42    (5) 

+ 
+ 
+ 

4  .V  =  5. 
2^  =  21. 

3x  +  2^  =  21 

15  +  2?/  =  21 
^  =  3 

(2) 

(8) 
(9) 

y  +  4z  =  7    (3) 

8a;  =  40    (6) 
x  =  5      (7) 

3  +  42  =  7  (10) 
.=  1(11) 

Equation  (1)  has  the  three  unknown  numbers,  x,  y,  z,  while  (2)  has 
only  X  and  y,  and  (3)  only  y  and  z.  We  therefore  combine  either  (2)  or 
(3)  with  (1)  so  as  to  eliminate  one  of  the  three  numbers.  We  select 
(3)  because  z  has  the  same  coefficient  in  both  (1)  and  (3).  Subtracting 
(3)  from  (1)  we  have  (4),  which  has  x  and  y,  the  same  unknown  num- 
bers as  are  in  (2).  Adding  twice  (2)  to  (4)  we  have  (6).  Dividing  (6) 
by  8  we  have  (7),  or  x  =  b.  Substituting  x  =  5  in  (2)  we  obtain 
(8),  which  reduced  gives  y  =  3.  Substituting  ?/  =  3  in  (3)  we  obtain 
(10),  which  reduced  gives  2=1. 

Hence,  for  solving  equations  containing  any  number  of 
unknown  numbers,  we  have  the  following 


From  the  given  equations  dedncc  equations  one  less  in  num- 
her,  eontaining  one  less  unknoivn  mtmher  ;  and  continue  thus 
to  eliminate  one  unknown  number  after  another,  until  one 
equation  is  obtained  containing  but  one  unknown  number. 
Reduce  this  last  equation  so  as  to  find  the  value  of  this  un- 
known number ;  then  substitute  this  value  in  an  equation 
containing  this  and  but  one  other  2inknown  number,  and, 
reducirig  the  resulting  equation,  find  the  value  of  this  second 


34 


ALGEBRA. 


unknoion  mwiher ;  substitute  again  these  values  in  an  equa- 
tion containing  no  more  than  these  two  and  one  other  un- 
known number,  and  reduce  as  befo7'e ;  and  so  continue,  till 
the  value  of  each  unknown  number  is  found. 

Note.  The  process  can  often  be  very  much  abridged  by  the  exercise  of 
judgment  in  selecting  the  unknown  number  to  be  eliminated,  the  equations 
from  which  the  other  equations  are  to  be  deduced,  the  method  of  elimination 
which  shall  be  used,  and  the  simplest  equations  in  which  to  substitute  the 
values  of  the  numbers  which  have  been  found. 


Find  the  values  of  tlu 
equations : 

(     X—     u  +  2z  =  l. 
3.  •]3ic-f27/-     z  =  ^. 
(4ic  —  3^+     2  =  3. 
rdx  +  2y=    z  +  12. 
'    ^     2/ +  32 -=33. 
-y    =% 
2tc  +  3?/  =  ]6. 

3  ?/  +  4  s  =  36. 

4  ar  +  5  ^  =  38. 


unknown  numbers  in  the  following 


(bx 


{  X         11         z 

4-      -4-  - 
^3^4 


3  +  4  +  5 


X         1/ 


=  56. 
=  43. 
=  35. 


Ans. 


Ans. 


Ans. 


Ans. 


(  ^  =  4. 

ra;  =  3. 

>  =  6. 
(s  =9. 

2. 

4. 

6. 


p  =  12. 
U  =  60. 
I  «  =  120. 


4   '   5   '   6 

f  X  +  ^  +  ^  +  ?t'  =  15.  \  x  =  2. 

I  y  +.+  ,,.  +  V  =  18.  1^  =  3. 

7.  }  X  -\-  z  -\-  in  4-  u  =  17.  Ans.   vj    2;  =  4. 

X  +  //  +  w  +  u  =  16.  u  =  5. 

[«  +  //+   '-  +  I'  =  14.  [  w  =  6. 

Note.  If  these  equations  are  added  together  and  the  sum  divided  by  4, 
we  shall  have  x  +  y-[-z-\-w-\-u  =  20;  and  if  from  this  the  given  equations 
are  successively  subtracted,  the  values  of  the  unknown  numbers  become 
known. 


K  z  -\-  X  =  b. 


ELIMINATION. 

1  x  + 

!/  +  ^' 

=  6. 

9. 

,v   +  ir 

=  9. 

n-+   z 

=  8. 

u  + 

U  +  i'^ 

=  7. 

35 


10. 


li>. 


13. 


|  +  y  =  l-2. 


+  ^  =  15. 


U  +  -9- 

la;       s; 

!'  +  '  =  - 

y+.       18 
114 

.  +  .-9- 


14.  { 


6  5 

-  +  - 

X  y 

X  z 


11 


10_ 

^  y 


15.  { 


X  ij  z  ?,' 
1112 
X       y       z       6 

111, 

-  4-  -  +  -  =  1. 
La;      y      z 


(  ^  +  A  //  +  ^  ^  =  10. 
W.\y^lx^yz^    S. 


Note.  Subtract  from  half  the 
sum  of  the  three  equation.s  each 
equation  successively. 


(a:  =  3. 

(;^=9. 


Note.     In   eliminating   do   not 
clear  of  fractions. 


16. 


X        z 

1_1 

X       y 


36  ALGEBRA. 


PROBLEMS 


PRODUCING    SIMPLE    EQUATIONS    CONTAINING    TWO    OR    MORE    UNKNOWN 
NUMBERS. 

27.  Many  of  the  problems  given  in  Chapter  III.  contain 
two  or  more  unknown  numbers ;  but  in  every  case  these  are 
so  related  to  one  another  that  if  one  becomes  known  the 
others  become  known  also;  and  therefore  the  problems  can 
be  solved  by  the  use  of  a  single  letter.  But  many  problems, 
on  account  of  the  complicated  conditions,  cannot  be  per- 
formed by  the  use  of  a  single  letter.  No  problem  can  be 
solved  unless  the  conditions  given  are  sufticient  to  form  as 
many  independent  equations  as  there  are  unknown  numbers. 

1.  A  farmer  sold  10  barrels  of  apples  and  3  barrels  of  pota- 
toes for  $2\)  ;  and  at  the  same  rate  4  barrels  of  apples  and 
5  barrels  of  potatoes  for  |2o.     Find  the  price  of  each  a  barrel. 

Let  X  =  the  no.  of  dollars  for  a  barrel  of  apples, 

and  y  =        "  "  "  "        potatoes. 

Then,  by  the  conditions,  10. r  +  3y  =  29, 
and  4x  +  by  =  2S. 

Solving  these  equations  we  have 

X  —  2  and  ?/  =  3. 

2.  Find  two  numbers  such  that  the  greater  exceeds  three 
times  the  less  by  6,  and  three  times  the  greater  exceeds  the 
less  by  22.  Ans.     5  and  9. 

3.  If  the  numerator  of  a  fraction  is  one  less,  the  resulting 
fraction  is  ^;  but  if  tlie  denominator  is  two  less,  the  resulting 
fraction  is  i.     Find  the  fraction. 

Let  -  =  the  fraction. 

y 

mi-  X—  11  ,         X  I 

Then  =  -  and  ■=  -  . 

y  3  2/-2       2 

Solving  these  eiiuations,  we  have 

X  =  5        and  y  —  12, 
and  the  fraction  is  -^^^ 


PROBLEMS.  37 

4.  Find  two  numbers  such  that  three  times  the  first,  minus 
four  times  the  second,  is  zero ;  and  half  the  first,  plus  a  third  of 
the  second,  is  9.  Ans.     12  and  9. 

5.  The  ages  of  two  persons,  A  and  B,  are  such  that  4  years 
ago  A's  age  was  three  times  B's,  and  10  years  hence  A's  age 
will  be  double  B's.      What  is  the  age  of  each  ? 

Ans.     A's  46 ;  B's  18. 

6.  Find  a  fraction  such  that  if  4  is  added  to  its  numerator 
the  resulting  fraction  is  |;  but  if  4  is  added  to  the  denominator 
the  resulting  fraction  is  equal  to  ].  Ans.     ^. 

7.  A  uvmiber  of  two  figures,  whose  sum  is  8,  is  such  that  if  36 
is  subtracted  from  it  the  order  of  the  figures  is  reversed.  What 
is  the  number  ?  Ans.    62. 

Let  X  =  the  tigure  in  the  tens'  place, 

and  1/  =        "  "       units'  place. 

Then  10  x  +  y  =  the  number. 

Then  lUz  +  ;/ +  36  =  10^  +  x 

and  z  +  //  =  S 

Solving  these  equations  we  have       a;  =  f5,         and  ij  =  2. 

8.  A  number  of  two  figures,  whose  sum  is  9,  is  such  that  if  27 
is  added  to  it  the  order  of  the  figures  is  reversed.  What  is  the 
number  ? 

9.  A  man  worked  9  days  and  his  son  5  days,  and  they  received 
for  their  work  ^23.  At  another  time  the  man  worked  7  days 
and  his  son  5  days,  and  they  received  $19.  What  were  the 
wages  of  each  ? 

10.  A  farmer  who  had  155  in  his  pocket  gave  to  each  man 
of  his  laborers  $3,  and  to  each  boy  |1,  and  had  18  left.  If  he 
had  given  each  man  13.50,  and  then  each  boy  $2,  until  he  had 
given  awa}'^  all  the  money,  2  boys  would  have  received  nothing. 
How  many  men  and  how  many  boys  did  he  hire  ? 


38  ALGEBRA. 

11.  As  William  and  Heury  were  talking  of  their  money, 
William  said  to  Henry,  "  Give  me  10  cents  and  I  shall  have 
5  times  as  much  as  you  will  have  left."  Henry  said  to  Wil- 
liam, "  Give  me  10  cents  and  I  shall  have  as  much  as  you  will 
have  left."     How  many  cents  did  each  have  ? 

Ans.     William,  40  cents ;  Henry,  20  cents. 

12.  A  and  B  began  business  with  different  sums  of  money. 
The  first  year  A  gains  $250,  and  B  loses  -fl50,  and  then  A's 
money  is  four-fifths  of  B's.  If  A  had  lost  -SISO,  and  B  gained 
1250,  A's  money  would  then  have  been  one-half  of  B's.  With 
what  sum  did  each  begin  ?  Ans.    A  11350  ;  B  $2150. 

13.  If  5  is  added  to  the  numerator  of  a  certain  fraction,  the 
value  will  be  a  unit ;  and  if  3  is  subtracted  from  its  denominator, 
the  value  will  be  A.     What  is  the  fraction  ? 

14.  Find  three  fractions  such  that  the  sum  of  the  first  and 
second  is  2^0,  of  the  second  and  the  third  -^-J,  and  of  the  first 
and  third  jV- 

15.  There  is  a  number  of  three  figures  whose  sum  is  6 ;  the 
right-hand  figure  is  equal  to  the  sum  of  the  other  two,  and  if 
198  is  added  to  the  number  the  order  of  the  figures  is  reversed. 
Find  the  number.  Ans.    123. 

Let  X  —  the  figure  in  units'  place. 

2/  =    "         "      "   tens'  place. 
z  —    "         "      "    hundreds'  place. 
Then  100 .-j  +  10.'/  +  x  =  the  number. 

By  the  conditions      x  +  y  +  z  =  6 
y  +  z  =  x 
and      100  2;  +  10  ?/  +  x  +  198  =  100  a;  +  10  /y  +  z. 
Solving  these  equations  we  have  x  =  Z,        //  =  2,         z—\. 

16.  Find  three  numbers  such  that  the  sum  of  the  first  and 
second  is  85,  of  the  second  and  third  142,  of  the  third  and 
first  87. 


PROBLEMS.  39 

17-  If  I  can  row  6  miles  down  a  river  in  an  hour,  while  it 
takes  me  3  hours  to  row  back,  at  what  rate  can  I  row  in  still 
water,  and  at  what  rate  does  the  water  flow  in  the  river  ? 

Let  X  -  no.  miles  an  hour  in  still  water, 

and  tj  —         "  "         the  water  flows. 

Then  by  the  conditions  a;  +  ?/ =  6  (0 

X  -  y  =  f  =  2  {.±) 

Add  (1)  and  (2)  2  2-  =  8 

Subtract  (2)  from  (1)  2  ?/  =  4 

a;  =  4 
y  =  2 

18.  A  father  wished  to  give  to  each  of  his  children  25  cents, 
but  found  he  needed  10  cents  more  than  he  had  to  do  it  ;  so  he 
gave  each  20  cents,  and  had  25  cents  left.  How  many  children, 
and  how  much  money  did  he  have  ?    Ans.  7  children,  and  11.65. 

19.  The  year  A.  d,  of  the  invention  of  printing  by  Gutenburg 
i.s  expressed  by  a  number  of  four  figures  whose  sum  is  14. 
The  units'  figure  is  double  the  tens',  the  thousands'  figure  is 
equal  to  the  hundreds  minus  the  tens  ;  and  if  4905  is  added  to 
the  number,  the  order  of  the  figures  will  be  reversed.  What  is 
the  date  ?  Ans.    1436. 

20.  A  boy  paid  80  cents  with  11  pieces  of  silver,  5  and  10 
cent  pieces.     How  many  pieces  of  each  did  he  give  ? 

21.  Peter  said  to  John:  ''If  you  give  me  fl,  I  shall  have 
twice  as  much  as  you  have  left."  John  said  to  Peter:  "If  you 
give  me  -f  1,  I  shall  have  three  times  as  much  as  you  have  now." 
H(jw  nnich  did  each  have  ? 

22.  A  certain  fraction  becomes  f  when  its  numerator  is  in- 
creased by  3,  and  its  denominator  is  multiplied  by  3;  but  if  its 
denomiijator  is  increased  by  3,  and  its  numerator  multiplied  by 
.3,  it  becomes  Yo-      Pind  the  fraction. 

23.  A  certain  fraction  becomes  3-  by  adding  1  to  both  numer- 
ator and  denominator,  but  \  by  subtracting  1  from  both  terms. 
Find  the  fraction. 


40  ALGEBRA. 

24.  A  certain  fraction  is  doubled  by  adding  6  to  its  numer- 
ator, and  9  to  its  denominator;  and  trebled  by  adding  2  to  its 
numerator,  and  taking  3  from  its  denominator.    Find  the  fraction. 

25.  A  and  B  can  do  a  piece  of  work  together  in  li  days. 
A  and  C  in  Ij  days,  and  B  and  C  in  If  days.  Find  the  time 
in  which  each  can  do  it  alone. 

26.  A  and  B  can  do  a  piece  of  work  together  in  3f  days.  It 
can  also  be  done  if  A  works  5  days,  and  B  3  days.  Find  the 
time  in  which  each  can  do  it  alone. 

27.  A  and  B  can  together  do  a  piece  of  work  in  20  days. 
They  work  together  15  days,  and  then  A  finishes  the  work 
alone  in  15  days  more.  In  what  time  can  each  do  the  work 
alone  ? 

28.  A  and  B  together  finished  a  piece  of  work  in  2^  days. 
If  A  had  worked  twice  as  fast,  and  B  half  as  fast,  it  would  have 
taken  them  1§  days.  In  liow  many  days  could  each  do  the 
work  alone  ? 

29.  The  sum  of  the  three  figures  of  a  number  is  9;  the  tens' 
figure  is  1 ;  and  if  the  order  of  the  figures  is  reversed  the  num- 
ber so  formed  exceeds  the  original  number  by  396.  What  is 
the  number  ? 

30.  If  I  divide  a  certain  number  of  two  figures  by  the  sum 
of  its  figures,  the  quotient  is  7,  and  the  remainder  6.  If  I 
reverse  the  order  of  the  figures,  and  divide  the  resulting  number 
by  the  difference  of  the  figures,  the  quotient  is  7,  and  the 
remainder  3.     Find  the  number. 

31.  A  number  of  three  figures,  whose  sum  is  12,  has  the  right- 
hand  figure  zero.  If  the  left-hand  and  middle  figures  change 
places  the  number  is  increased  by  180.     Wliat  is  tlic  number? 

32.  A  man  has  1900  at  interest  in  three  sums,  the  first  at 
4%,  the  second  at  5%,  and  the  third  at  6%,  receiving  for 
interest  |47  a  year.  The  part  at  5%  is  lialf  as  much  as  the 
other  two  together.     Find  the  three  parts. 


ALGEBRAIC   NUMBERS.  41 


CHAPTER    Y. 


ALGEBRAIC    NUMBERS. 


28.  Algebra,  is  the  science  which  treats  of  numbers. 
These  numbers  are  represented  by  figures  and  letters.  In 
general  it  is  agreed  that  the  first  letters  of  the  alphabet,  a,  h,  c, 
etc.,  shall  stand  for  what  are  called  hioivn  numbers,  that  is, 
those  whose  values  are  given  ;  and  the  last  letters,  x,  y,  z, 
etc.,  for  unknoivn  numbers,  that  is,  those  whose  values  are 
to  he  determined. 

29.  In  Algebra  numbers  are  of  tivo  kmds,  positive  and 
negative.  Thus,  on  the  scale  of  the  thermometer,  the  num- 
bers above  the  zero  mark  are  called  positive,  before  which  no 
sign,  or  the  +  sign,  is  placed ;  and  those  hcloif  the  zero  mark 
a,re  called  negative,  before  which  the  —  sign  is  placed.  80°, 
or  +  80°,  indicates  eighty  degrees  cd)ove  zero,  and  —  12°  indi- 
cates twelve  degrees  below  zero. 

So  the  numbers  representing  north  latitude,  cast  longi- 
tude, gain,  etc.,  are  usually  called  positive,  and  their  oppo- 
sites,  that  is,  south  latitude,  west  longitude,  loss,  etc.,  are 
called  negative.  Negative  numbers,  as  compared  with  posi- 
tive numbers,  mean  opposite  in  direction,  or  in  effect,  and  are 
as  real  as  positive  numbers.  There  is  nothing  in  the  nature 
of  things  to  prev^ent  our  considering  positive  numbers  as 
negative,  and  negative  numbers  as  positive. 


42  ALGEBRA. 

30.  A  clear  understanding  of  these  numbers  in  all  their 
relations  can  best  be  obtained  through  the  device  of  a  straight 
line  with  the  zero  point  at  its  centre,  and  positive  numbers 
extending  to  the  right,  and  negative  numbers  to  the  left, 
indefinitely,  thus, 

.  .  .  _5,  _4,  _3,-  2,   -1,      0,   +1,  +2,   +3,  +4,   +5,  .  .  . 

. — i 1 1 h- 1 1 1 1 1 \ 1 — 

a  series,  embracing  all  possible  integral  numbers,  which 
increase  by  one  indefinitely  from  left  to  riglit,  and  decrease 
by  one  indefinitely  from  right  to  left.  In  this  series  every 
negative  number  is  considered  to  be  less  than  zero ;  and,  in 
general,  every  number  in  the  series  is  considered  to  be  less 
than  any  number  following  it,  and  greater  than  any  number 
preceding  it,  that  is,  relatively  so. 

In  this  relative  sense  the  phrases  greater  tlmii,  less  than, 
less  than  zero,  are  to  be  understood,  unless  the  contrary  is 
expressed. 

Arithmetic  takes  into  account  that  part  only  of  the  series 
to  the  right  of  the  zero,  while  algebra  makes  use  of  the  whole 
series.  From  this  it  will  be  seen  at  once,  that,  in  algebra, 
addition,  subtraction,  multiplication,  and  division,  covering 
as  they  do  both  positive  and  negative  numbers,  must  have 
a  more  extended  signification  than  in  arithmetic. 


ADDITION.  43 


CHAPTEE    VI. 
ADDITION. 

31.  Addition  in  algebra  is  the  process  of  finding  the 
aggregate,  or  sum,   of  two  or  more  algebraic  numbers. 

If  the  numbers  to  be  added  are  all  positive,  the  algebraic 
sum,  is  positive  and  equal  in  amount  to  the  number  of  posi- 
tive units ;  if  they  are  all  negative,  the  algebraic  sum  is 
negative  and  equal  in  amount  to  the  number  of  negative 
units  ;  and  if  they  are  both  positive  and  negative,  the  alge- 
braic sum  is  positive  if  the  positive  units  are  in  the  excess, 
and  is  equal  in  amount  to  that  excess,  negative  if  the  nega- 
tive units  are  in  the  excess,  and  is  equal  in  amount  to  that 
excess,  and  zero  if  the  sums  of  the  positive  and  negative 
units  are  equal.     Thus, 


(A) 


32.    To  illustrate  (1)  in  the  group  above,  move  a  pencil  from 
the  zero  point  along  the  line  of  numbers  in  the  positive  direction 

fi^°  +  — =@a 

-5,   -4,   -3,   -2,   -1,      0,   +1,   +2,   +3,   +4,   +5, 

H 1 1 \ 1 1 1 1 1 H H 

(that  is  to  the  right)  five  spaces,  then  three  spaces ;  the  dis- 
tance from  the  starting  point  is  eight  spaces  to  the  right,  and  is 
represented  by  +  8  s,  the  algebraic  sum  of  the  distances  moved. 
To  ilhistrate  (2),  move  a  pencil  from  the  zero  point  along  the 
Hne  of   numbers   in   the  negative  direction  (that   is  to  the  left) 


(1) 

+  5+  (+  3)  =  +  8 

(2) 

-5+  (-3)  =.-8 

(3) 

+  5  +  (-  3)  =  +  2 

(4) 

-5+  (+3)  =-2 

(5) 

+  5+  (-5)=      0 

44  ALGEBRA. 

five  spaces,  theu  three  spaces;  the  distance  from  the  starting 
point  is  eight  spaces  to  the  left,  and  is  represented  by  —8  s,  the 
algebraic  sum  of  the  distances  moved. 

-5,  -4,   -3,  -2,  -1, 0,   +1,   +2,   +3,   +4,   +5, 

To  illustrate  (3),  move  a  pencil  from  the  zero  point  along  the 
line  of  numbers  in  the  positive  direction  five  spaces,  then  in  the 
negative  direction  three  spaces ;  the  distance  from  the  starting 
point  is  two  spaces  to  the  right,  and  is  represented  by  +2  5,  the 
algebraic  sum  of  the  distances  moved. 

To  illustrate  (4),  move  a  pencil  from  the  zero  point  along  the 
line  of  numbers  in  the  negative  direction  five  spaces,  then  in  the 
positive  direction  three  spaces ;  the  distance  from  the  starting 
point  is  two  spaces  to  the  left,  and  is  represented  by  —  2  s,  the 
algebraic  sum  of  the  distances  moved. 

To  illustrate  (5),  move  a  pencil  from  the  zero  point  along  the 
line  of  numbers  in  the  positive  direction  five  spaces,  then  in  the 
negative  direction  five  spaces;  the  distance  from  the  starting 
point  is  zero,  and  is  represented  by  0  s,  the  algebraic  sum  of  the 
distances  moved. 

33.  The  algebraic  sum  is  not  then,  as  in  arithmetic,  the 
entire  number  of  spaces  moved,  but  the  distance  at  the  ces- 
sation of  the  movement,  from  the  starting  point. 

And,  in  general,  the  algebraic  sum  of  several  numbers  is 
the  deviation  of  the  result  from  zero,  the  positive  units  being 
counted  on,  or  employed  to  affect  tlie  result,  according  to 
their  number,  in  one  way,  and  the  negative  units  being 
counted  off,  or  employed  to  affect  the  result,  according  to 
their  number,  in  the  opposite  v^ay. 

In  a  parliamentary  body  the  members  on  one  side  may  be 
supposed  to  represent  positive  numbers,  and  the  members  of 
the  opposition  negative  numbers.  With  such  a  supposition 
the  algebraic  sum  is  the  majority.  Thus,  if  there  are  87  on 
one  side,  and  69  in  opposition,  the  majority  is  87  —  69,  or  IS  ; 


ADDITION.  45 

that  is,  the  majority  is  the  sum  of  87  and  —  69,  Pairing 
does  not  affect  the  result.  Thus,  if  in  the  case  above  2  on 
one  side  had  paired  with  2  in  opposition,  the  numbers  would 
stand  85  and  —  67,  and  the  majority  would  have  been 
85  —  67,  or  18,  as  before. 

34.     Illustrative   Problems. 

1.  Suppose  a  man  to  walk  along  a  straight  road  70  rods  for- 
ward and  then  50  rods  backward,  his  distance  from  his  starting 
point  is  20  rods. 

But  if  he  first  walks  50  rods  forward,  and  then  70  rods  back- 
ward, his  distance  from  his  starting  point  would  be  20  rods, 
but  on  the  opposite  side  of  his  starting  lioint. 

The  corresponding  algebraic  statements  are 

70  rd.  +  (-  50  rd.)  =  +  20  rd. 
50  rd.  -f  (-  70  rd.)  =  -  20  rd. 

2.  Suppose  that  I  have  a  house  worth  $5000,  and  a  piece  of 
land  worth  $2000,  and  that  I  owe  $500;  then  the  net  value  of 
my  property  is  $5000  +  $2000  +  (-  $500)  =  -f  $6500.  Again, 
suppose  my  house  is  worth  $5000,  my  land  $2000,  while  I  owe 
$8000;  then  the  net  value  of  my  property  is  $5000  -f  $2000 
+  (_  18000)  =  —  11000  ;  that  is,  I  am  worth  —  $1000,  or,  in 
other  words,  I  owe  $1000  more  than  I  can  pa}'. 

3.  A  merchant  was  engaged  in  business  for  two  years ;  the 
first  year  he  gained  $2000,  and  the  second  he  gained  —  $500 
(that  is,  he  lost  $500).  How  did  he  come  out  at  the  end  of  the 
second  year  ?  Ans.    With  a  gain  of  $1500. 

4.  A  thermometer  indicated  -f  50*^  (50°  above  0)  ;  it  then 
fell  20°,  then  rose  40°.  What  temperature  did  it  then  indicate  ? 
Had  it  fallen  40°,  instead  of  risen  40°,  what  would  have  been 
the  temperature?  Ans.    70°;  —10°. 

5.  A  balloon  ascending  from  Boston  was  driven  due  east 
30  miles  the  first  hour,  and  25  tlie  second,  wlieu,  rising  liigber, 


46  ALGEBRA. 

it  encountered  a  wind  which  swept  it  due  west  at  the  rate  of 
40  miles  an  hour.  Required,  its  position  at  the  end  of  4  hours 
from  the  start. 

Ans.    25  miles  west  of  Boston;  or,  —25  miles  from  Boston. 

In  this  example  which  contains  the  greater  number  of  units, 
the  arithmetical  or  the  algebraic  sum  ? 

Ans.     The  arithmetical,  b3^  110  units. 

6.  A  ship  sailing  at  the  rate  of  6  miles  an  hour,  sails  three 
hours  due  east  and  one  hour  due  w^est.  How  far  is  she  from  her 
starting  point  ? 

7  If  a  ship  sails  at  the  rate  of  10  miles  an  hour  three  hovirs 
due  east,  and  four  hours  due  west,  how  far  is  she  from  her 
starting  point  ? 

8.  A  man  earns  $3  a  da^^,  and  runs  up  a  store-bill  of  $4  a 
day.     Write  down  his  standing  at  the  end  of  6  da^'s. 

DEFINITIONS. 

35.  An  Algebraic  Expression  is  a  single  number,  or  a  collec- 
tion of  numbers,  generally  connected  by  algebraic  signs. 

36.  A  Monomial  is  an  algebraic  expression  which  contains 
a  single  term ;  as,  a,  or  4  x,  or  7  c xz. 

37.  A  Polynomial  is  an  algebraic  expression  which  con- 
tains two  or  more  terms  ;  as,  «  -I-  h,  5  x  +  o  y  —  6  h  x  y, 
or  5  +  3  c  -  4  r/  +  r. 

38.  A  Binomial  is  a  polynomial  of  two  terms  ;  as,  2  «  -|-  5  y, 

or  a  —  h. 

39.  A  Trinomial  is  a  polynomial  of  three  terms ;  as, 
«  4-  5  .y  —  7  a  h. 

40.  Like  Terms,  or  Similar  Terms,  are  those  which  do  not 
differ,  or  differ  only  in  their  signs  or  coefficients  ;  as,  5  a  h, 
and  —3  ah.  Other  terms  are  unlike  or  dissimilar ;  as,  1  xy 
and  n  h  y. 


ADDITION.  47 


ADDITION  OF  ALGEBRAIC  LITERAL  EXPRESSIONS. 

Case   I. 

41.    To  find  the  Sum  of  Monomials  when  they  are  Similar  and 
have  Like  Signs. 

1.    William  has  7  apples,  James  5  apples,  and  Henry  4  apples  ; 
how  many  apples  have  they  all  ? 
7  ajiples. 


5  apples, 
4  apples, 


It   is   evident  that  just  as 
\       i  a  '' 

or  letting  a  \       rj  ^  7   apples  and   5  apples   and 

represent    ■{  4  apples  added  together  make 

one  apple,    j    16  apples,  so  7  a  and  5  a  and 

16  apples,    j  I    16  «  4  a  added  together  make  16  a. 

In  the  same  way  —  7  «  and  —  5  a  and  —  4  a  are  equal  together 
to  —  16  a. 

Therefore,  when  the  monomials  are  similar,  and  have  like 
signs,  we  have  the  following 

Rule. 

Add  the  coefficients,  and  to  their  sum  annex  the  common 
letter  or  letters,  and  prefix  the  common  sign. 


(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

(7.) 

2by 

Qxy 

3az 

4a 

-    6z 

—  6  ax 

5hy 

2xy 

az 

a 

-   5z 

—  5  ax 

Sby 

5xy 

1  az 

la 

-    dz 

—     ax 

Ihy 

Txy 

2az 
13  a  z 

2a 

—       z 

—  4  ax 

17  by 

-21  z 

8. 

What  is  the 

sum  of   X, 

bx,  7x, 

and  2x? 

Ans.    15  a;. 

9.    What  is  the  sum  oi  3  b  x,   6  bx,   5  b  x,   and  2b  x  ? 

10.  What  is  the  sum  of  5  ax,   6  ax,   11  ax,   and  Sax? 

11.  What  is  the  sum  oi  —xy,  —Ixy,  —2xy,  and  —  xy? 

Ans.   —  11  xy. 

12.  What  is  the  sum  of  —2  be,  —bbc,  —Abe,  and  — 6  be? 

13.  What  is  the  sum  of  —  2  b  c  d,    —  5bcd,   and  —3bcd? 


48 


algp:bra. 


Case    II. 

42.    To  find  the  Sum  of  Monomials  when  they  are  Similar  and 
have   Unlike  Signs. 

1.  A  man  earns  9  dollars  one  week,  and  the  next  week  earns 
nothing  and  spends  4  dollars,  and  the  next  week  earns  7  dollars, 
and  the  fourth  week  earns  nothing  and  spends  3  dollars  ;  how 
much  money  has  he  left  at  the  end  of  the  fourth  week  ? 

If  what  he  earns  is  indicated  by  +,  then  what  he  spends  will 
be  indicated  by  — ,  and  the  example  will  ap^jear  as  follows: 


+  9  dollars, 

-  4  dollars, 
+  7  dollars, 

—  3  dollars, 


\  +  dd 

or,  letting (/  |  —  ^d 

represent    -{  -{-  7  d 

one  dollar,  —  '3d 

I  +  9d 


Earning  9  dollars  and 
then  spending  4  dollars, 


the   man   would  have  5 
dollars  left ;  then   earn- 
ing 7  dollars,  he  would 
+  9  dollars,    J  I    +  9 1?  have    12  dollars  ;     then 

spending  3  dollars,  he 
woidd  have  left  9  dollars.  Or  he  earns  in  all  9  dollars  +  7  dollars 
=  16  dollars;  and  spends  4  dollars  +  3  dollars  =  7  dollars  ;  and  there- 
fore has  left  the  difference  between  16  dollars  and  7  dollars  =  9  dollars ; 
hence  the  sum  of  +  9  d ,  —  4  d,  +  7  d,  and  —  3  rf,  is  -f-  9  rf. 

Therefore,  when  the  terms  are  similar,  and  have  unlike  signs, 
we  have  the  following 

Rule. 

Find  the  difference  hetimen  the  sum  of  the  coefficients  of  the 
2Jositive  terms,  and  the  sum  of  the  coefficients  of  the  negative 
terms,  and  to  this  difference  annex  the  common  letter  or  letters, 
and  prefix  the  sign  of  the  greater  sum. 


(2.) 

(3.) 

(4.) 

(5.) 

X 

X 

lb  ac 

12  X7JZ 

3x 

-    3x 

6  ac 

—  14:xyz 

—       X 

7  X 

—    7  a  c 

xyz 

Ix 

-    5x 

—        ac 

-12  xyz 

-4.x 

13  a; 

Sac 

15  xyz 

Q)X 


IQ  ac 


' 

ADDITION. 

(6.) 

(7.) 

2hhcd 

S  (a  -h  b) 

-50  1)0  d 

-7  (a  +  b) 

10  bed 

5  {a  +  b) 

-48  bed 

-  3  {a  +  b) 

1  bed 

-2(a  +  b) 

49 


-56  bed  (a  +  b) 

8.  Find  the  sura  of  7  a:,   —  14  x,    13  x,   and  —  3  x. 

9.  Find  the  sum  of  8  (a  +  b),   11  (a  +  b),  —  {a  +  b),  and 
4  (a  +  l>).  Ans.    22  (a  +  6). 

10.  Find  the  sum  of   —  5  a  z,    +  5  a  z,   +  10  as,   +25  a 2;, 
and  —  13  a  z. 

11.  Find   the   sum   of   27  b  c,     -33  be,      -Ubc,     28  be, 
and  -12  be. 

12.  Find  the  sum  of  —11  be,   be,   11  be,  -be,   13 be,   and 
-be. 

13.  Find  the  sum  of    (b  +  c),    —  (b  +  c),    12  (6  +  c),    and 
-  10  (6  +  c).  Ans.   2  (5  +  c). 

Case    III. 
43.     To  find  the  Sum  of  any  Monomials. 

From  (3)  in  group  (A)  page  43,  it  follows  that 

a  +  (—  b)  =  a  —  b 

and  that  the  sum  of  a,  —  b,  and  —  c,  or 

a  -{-  (—  b)  +  (—  e)  =  a  —  b  —  e 

No   further   reduction  is  possible,  and   therefore,  to  add 
dissimilar  monomials  we  have  the  following 

Rule. 

Write  them  one  after  the  other,  each  with  its  proper  sign. 

All   algebraic   expressions    can   be   so    written,    and    the 
result,   without   further   reduction,  is   sometimes  called   an 
jraic   sum. 

4 


50  ALGEBRA. 

44.  It  should  be  remarked  that 

a-\-b-{-c  =  a-i-c-\-b=zb  +  a-\-c—  etc. 
that  is,  the  sum  of  any  number  of  algebraic  expressions  is 
independent  of  their  order. 

45.  Further,  +(^a  +  b)^  +  a-\-b 

For,  to  put  a  and  h  together,  and  then  add  the  result  to 
what  has  gone  before,  is  the  same  as  to  add  both  a  and  h  to 
what  has  gone  before.     Similarly, 

a  +  ib  +  c)  =  {a-\-b)  +  c 

that  is,  the  sum  of  any  number  of  algebraic  expressions  is 
independent  of  the  mode  of  groiqnng  them. 

46.  It  follows  that 

the  sum  of  4  a  and  5  &  is  neither  9  a  nor  9  b,  and  can  only  be 

expressed  in  the  form  of4a  +  5i^,  or5i  +  4a;  and  the  sum  of 

7  a  and  —Sbi&7a—Zb,o\i—^b  +  la.    In  finding  the  sum  of 

4  a,  5  b,  7  a,  and  —3  b,  the  a's  can  be  added  together  by  Case  I., 

and  the  b's  by  Case  II.,  and  the  two  results  arranged  according 

to  §§  44,  45 ;    thus,  4:a  +  5b+7a-3b  =  4ta-{-7a  +  5b 

—  3b  =  lla  +  2b,  or  2  5  +  11  a,  regardless  of  the  order  of  the 

terms. 

1.    Find  the  sum  oi6b,  —  3c,  5  a,  4  ?/,  3  a,  —5  c,  Tab,  Ab, 

Qa,  8  c,  —  4y,  and  —  4  a  5. 

For  convenience,  similar  terms 
6b-3c+    oa  +  4.y+7ab      ^^^  ^^^^.^^^^^^  ^^^^^^^  ^^^^  ^^^^^ .  ^j^^^ 

**~o^+     3a— 42/— 4a&      by  Case  I.  the  first  column  at  the 
left  is  added  ;  the  second  by  Case 


+  8  c  +    6  a 


10  b  +14:  a  +3ab      II.,  and  so  on. 

Therefore,  to  find  the  sum  of  any  monomials,  we  have  the 
following 

Rule. 

Add  the  similar  terms  according  to  Cases  I.  and  If.,  and 
write  after  these  results,  in  any  order,  the  dissimilar  ones, 
each  with  its  proper  sign. 


ADDITION.  51 

2.  Find  the  sum  of  Ga,  —  5  6,  -\-  oc,  -{■  -ib,  —  7  c,  -{-  6d, 
—  3  c,  +  8  a,   and  —  o  d. 

3.  9  +  (-  7)  +  (-  5)  +  0  +  (+  8)  +  (-  4)  =  ? 

4.  Find  the  sum  of  5  a,  —  6  0,  +  -ix,  +  ib,  —  3  s,  +5  x, 
+  a,  +  3 b,  —  4 z,  ■}-  oj/,  —5a,  +  Sb,  —a,  —  4 c,  —7x,  and 
+  8^. 

5.  3x+(-5y)  +  {-\-6x)  +  (+ 8y)  +(-4y)=? 

6.  4.a  +  {-5b)  +  {+6a)-{-2b)  +  (-Sa)-^(-^7b)  =  ? 

Case  IV. 
47.     To  find  the  Sum  of  Polynomials. 

Polynomials  are  groups  of  monomials,  and  hence  every- 
thing necessary  to  a  complete  understanding  of  their  addition 
has  been  explained  in  the  three  foregoing  cases. 

We  have,  therefore,  for  the  addition  of  polynomials  the 
following 

Rule. 

Write  similar  terms  imdcr  each  other,  find  the  sum  of  eaeh 
column,  and  connect  the  several  sums  -with  their  proper  signs. 

1.    Find  the  sum  of 

3a  +  2b-c,  -a-^3b  +  2c,  2a-3b-{-5c. 
3a  +  2b-c 

—  a  +  3b  +  2c 
2a-3b  +  5c 


4ct  +  26  +  6c 
2.    Add 
—  5ab-\-Sbc—Qac,  Sab  —  ibc  +  3  ac,  -2  ab-2bc  ^2<f,c. 

—  5aJ  +  86c  —  9ac 

Sab  —  4:bc  -\-3ac 

-2ab-2bc  +  2aG 

ab  +  2bc  —  -iac 


52  ALGEBRA. 

3.    Add    2x-  Sax  +  1,   ox  —  a,r  +  5,  x  -{-  2ax  —  S. 

2x-3ax  +  1 

5a;  —     ax  -{-  5 

X  +  2ax  —  8 

Sx-2ax-2 

Add  the  following  : 

4.  4a  +  3^  +  5c,    -2a  +  3b  — 8c,    n  —  b  +  c. 

Alls.    ,'}(/  +56  —  2c. 

5.  5  a  — 2  b  — 6  c,    S  a  —  o  b  -\-  2  c,    —  2  a  +  3  b  -  c. 

Alls.     6a  —  46  —  5  c. 

6.  -15a- 196- 18  c,    14  a  +  15  6  +  8  c,   a +  56  + 9c. 

Ans.     b  —  c. 

7.  25  a  — 15  b  i-  c,    13  a  —  10  6  +  4  c,   «  +  20  6  —  c. 

Ans.     39  a  —  5  6  +  4  c. 

8.  4  a  +  2  6  +  8,     -  2  «  -  7  6  -  13,     -  7  a  -  15  6  -  7, 
3  a  +  4  6  -  11. 

9.  2  a  -  3  6  +  c,    15  a  -  21  6  -  8  c,    3  a  +  24  6  +  7  c. 

Ans.     20  a. 

10.  23a  — 176 -2c,   —9a +  156  + 7c,    —13a +  36  — 4c. 

Ans.     a  +  6  +  c. 

11.  a  +  2  6  -  c,    2  a  -  6  +  2  c,    _  3  a  -  6  -  c. 

12.  2  a  -  5  6  +  2  c,  26  -  5  c  +  2  a,  2  c  -  5  a  +  2  6. 

Ans.     —a  —  b  —  c. 

13.  3  ^.  +  7  -  5  .r,  2  a;  -  8  -  9  a,  4  a  -  2  a;  +  3  sc. 

Ans.     cc  —  5  a  —  1. 

14.  5  a  -  3  c  +  ^^,    6  -  2  a  +  3  c?,   4  c  -  2  a  -  3  o?. 

Ans.     a  +  6  +  c  +  f/. 

15.  5a-3a6  +  9a6  + 176,    -  2a  +  6a6  -  4  a6  -  126, 
h  —  ^ah  —  5ah  —  a,    2ab  — 2a  — 6b  —  ab. 

16.  a  —  4a6  +  6a6c,    a6  —  10a6c  +  c,    6  +  3a6  +  a6c. 

Ans.     a  +  6  +  c  —  3a6c. 

17.  6a  +  76  +  llc  +  4c?  +  9,    8c  -  5  6  +  2  a  -  11  </ -  T, 
3a-5b-9c-\-e,  d-\-a  +  10b-3c,  -7 a-6b-7c-5d-10. 

Ans.     5a +  b -lid -2. 


SUBTRACTION.  53 


CHAPTER    VII. 


SUBTRACTION. 

48.  SuBxrtACTiON  is  the  process  of  finding  the  difference  be- 
tween two  numbers.  This  difference  is  the  number  of  units 
which  lie  between  the  two  numbers,  or  is  what  must  be 
added  to  the  subtrahend  to  produce  the  minuend. 

Subtrahend  +  Remainder  —  Minuend. 

49.  Thus,  in  the  group  below,  considering  the  5's  as  minu- 
ends and  the  3's  as  subtrahends,  by  determining  what  must 
be  added  to  the  subtrahend  to  produce  the  minuend,  that  is, 
by  addition,  we  obtain  the  following : 


(1)  +  5  -  (-  3)  -  -h  8 

(2)  -5- (+3)  =-8 

(3)  +  5  -  (+  3)  =  +  2 

(4)  _5-(-3)=-2 


(B) 


Wherever  the  points  indicated  by  the  minuend  and  sub- 
trahend are  situated  in  the  series  of  numbers,  a  certain 
number  of  spaces  must  be  between  them,  and  this  number 
with  the  appropriate  sign  represents  their  algebraic  difference. 
If  the  movement  is  to  the  right  in  going  from  the  point  indi- 
cated by  the  subtrahend  to  the  point  indicated  by  the  minu- 
end, the  -t-  sign  must  be  prefixed ;  if  to  the  left,  the  —  sign. 

To  illustrate  (1),  in  the  group  above,  move  a  pencil  from  the 
point  three  spaces  to  the  left  of  0  to  a  point  five  spaces  to  the 

s®^  +  — --^a 

-5,    -4,    -3,    -2,   -1,       0,    +1,    +2,   -^3,    +4,    -1-5, 
1— 1 h 1 1 ; 1- 1 1 1 1 

right  of  0 ;  the  distance  moved  is  eight  spaces  in  the  positive 
direction;  hence  the  algebraic  difference  is  -{-  8s. 


54  ALGEBRA. 

To  illustrate  (2),  move  a  pencil  from  the  point  three  spaces  to 
the  right  of  0  to  a  point  five  spaces  to  the  left  of  0 ;  the  distance 
moved  is  eight  spaces  in  the  negative  direction  ;  hence,  the 
algebraic  dlffer'ence  is  —  8  s. 

fi@°  +  .  —^WSi 

_5,   -4,    -3    -2,    -1,       0,    +1,    +2,   +3,   +4,    +5. 


To  illustrate  (3),  move  a  pencil  from  the  point  three  spaces  to 
the  right  of  0  to  a  point  five  spaces  to  the  right ;  the  distance 
moved  is  two  spaces  in  the  positive  direction ;  hence,  the  alf/e- 
braic  difference  is  +  2  s. 

T''o  illustrate  (4),  move  a  pencil  from  a  point  three  spaces  to 
the  left  of  0  to  a  point  live  spaces  to  the  left  of  0  :  the  distance 
moved  is  two  spaces  in  the  negative  direction  ;  hence,  the  alge- 
braic difference  is  —  2  s. 

50.  From  groups  (B)  and  (A),  pp.  53,  43,  (2)  and  (1,)  it 
follows  that  subtracting  a  positive  number  is  equivalent  to 
adding  an  equal  negative  number,  and  subtracting  a  negative 
number  is  equivalent  to  adding  an  equal  positive  number. 

Therefore,  to  subtract  one  number  from  another,  we  have 
the  following 

Kule. 

Change  the  sign  of  the  subtrahend  and  proceed  as  in  addition. 

51.    Illustrative  Problems. 

1.  Suppose  I  am  worth  17000;  it  matters  not  whether  a 
thief  steals  13000  from  me,  or  a  rogue  having  the  authority 
involves  me  in  debt  13000  for  a  worthless  article ;  for  in  either 
case  I  shall  be  worth  only  $4000.  The  thief  subtracts  a  positive 
quantity  ;  the  rogue  adds  a  7iegative  quantity. 

The  corresponding  algebraic  statements  are 

$7000  -  (+  $3000)  =  +  $4000 
and  $7000  +  (—  $3000)  =  +  14000 


SUBTRACTION.  55 

2.  The  Roman  emperor,  Claudius  I.,  was  born  B.C.  10,  and 
died  A.  D.  54.     How  old  was  he  when  he  died  ? 

3.  If  A  has  1600  and  no  debts,  and  B  has  no  property  but 
owes  1200,  how  much  better  off  is  A  than  B  ?  Ans.     1800. 

4.  The  longitude  of  Berlin  is  13°  E.,  that  of  Boston  71°  W. 
What  is  their  difference  of  longitude? 

5.  The  longitude  of  St.  Louis  is  90°  W.,  that  of  Boston  71°  W. 
What  is  their  difference  of  longitude  ? 

Subtraction  of  Algebraic  Literal  Expressions. 

52.  From  group  (B)  we  learn  that,  in  general,  the  suh- 
iraction  of  a  positive  number  is  equivalent  to  adding  an  equal 
negative  numher,  and  the  siibtraction  of  a  negative  numher  is 
equivalent  to  aelding  an  equal  positive  numher: 

Therefore,  for  the  subtraction  of  monomials  and  polynomials, 
we  have  the  following 

Rule. 

Change  the  sign  of  eaeh  term  of  tlie  subtrahend  from  +  to  —, 
or  —  to  +,  or  suppose  each  to  be  changed,  and  then  proceed  as 
in  addition. 


(1-) 

(2.) 

(3.) 

(4.) 

(5.) 

(6.) 

('•) 

Min. 

9 

9 

9 

9 

9 

9 

9 

Sub. 

9 

6 

3 

0 

-3 

-6 

-9 

Rem. 

0^ 

3 

6 

"9 

12 

15 

18 

In  examples   1-7,    the  minuend  remaining  the  same  while  the 

subtrahend  becomes   in   each  3  less,  the  remainder  in   each  is 

3  greater  than  in  the  preceding. 

(8.)      •  (9.)        (10.)       (11.)       (12.)       (13.)       (14.) 
Min.        9  6  3  0-3-6-9 

Sub.         9^       9        9  9      9      9      9 

Rem.        0         -3        -6        ^^       -12       -15       -18 
In  examples  8-14,  the  minuend  in  each  becoming  3  less  while 

the  subtrahend  remains  the  same,  the  remainder  in  each  is  3  less 

than  in  the  preceding. 


56  ALGEBRA. 

(15.)  (16.)  (17.)      (18.)       (19.)       (20.)       (21.) 

Mill.           9  6  3           0-3-6-9 

Sub.            9  ^  j3          ^        -^        -6^       -9^ 

Rem.            0  0  0            0             0              0              0 


In  examples  15-21,  both  minuend  and  subtrahend  decreasing 

r  3,  the 

remainder  remains 

the  same. 

(1-)            (2.) 

(3.) 

(4.) 

(5.) 

Min. 

9  a            10a; 

-7y 

-12z 

llcfZ 

Sub. 

5  a         -  3  a- 

-3y 

+   7z 

3cd 

Rem. 

4a            13a; 
(6.) 

_4y 

-19z 

Min. 

5  a  +  3h  —  7  c 

4a-3b- 

5  c 

Sub. 

3a  -\-2b  —  5c 

6a              + 

5c-4:d 

Rem.         2a  +     b-2c  -  2 a  -  3(j  -  10c  +  4d 

8.  From  12  a  subtract  3  a. 

9.  From  —  17  a  ^  subtract  —  3  a  h. 

10.  From  10  a  —  6&  —  8c  take  4  a  +  2  ^^  -  3  c. 

11.  From  3a;  +  6^  +  ll2;  take  3y-3  z. 

12.  From  1  x-12y-12z  take  6  a:  -  8  y/  +  3  «. 

13.  From  3a^<  +  4^f/-5ac-6/>fZ  take  3ab  +  2cd-3ac-\-3bd. 

14.  Subtract  5  x  +  3  y  —  1  from  6  x,  and  add  tlie  result  to 
lx-5y-\-2. 

15.  AYhat    must    be    added    to    3  a  —  4  J  +  r    to    produce 
5a  +  3b -2c. 

16.  What  must  be  subtracted  from    10  a  —  3  ^^  —  2  c  +  f^  to 
leave   3  a  +  5b  —  c  —  d. 

17.  To  what  must   3  a  — 'lb  ^  r  bo  added  to  produce  zero  ? 

18.  What  must  be  added  to  1  a  —  A  b  -{-  11  to  give  11  a  —  8  ? 

19.  From    what    must    the    sum    of    8  <(  —  3,    3  ^^  -  4,    and 
4  a  +  6  be  subtracted  to  give   5  a  —  25  ■' 


SUBTRACTION.  57 

Removal  and  Introduction  of  Brackets. 

53.  The  addition  or  subtraction  of  a  polynomial  may  be 
indicated  by  enclosing  the  polynomial  in  a  bracket  and 
prefixing  the  sign  +  for  addition,  and  the  sign  —  for  sub- 
traction. The  polynomial  d  —  c+f  added  to  the  polynomial 
a  +  b  —  c,  that  is,  o.  +  b  —  c  -{-  {d  -  e  +  /),  equals  by  the 
mles  of  addition  a  +  b  —  c  +  d  —  c-\-f\  and  the  polynomial 
d  —  e  ■\-  f  subtracted  from  the  polynomial  a  +  b  —  c,  that 
is,  a  -{-  b  —  c  —  {d  —  c  -{■  /),  equals  by  the  rides  of  subtraction 
a  -{■  b  —  c  —  d  +  e  —f 

Therefore,  for  the  removal  of  brackets  we  have  the  fol- 
lowing 

Rule. 

Whe7i  the  bracket  with  the  plus  sign  before  it  is  removed, 
the  included  terms  must  be  rewritten  without  change  of  sign ; 
but  ivhen  a  bracket  vnth  the  minus  sign  before  it  is  removed, 
the  included  terms  must  be  rewritten  with  change  of  sign. 

54.  When  there  are  several  brackets,  they  may  be  removed 
one  at  a  time. 

Thus,  4x-{3cc-(2x-  a^^^a)] 

=  4  X  —  {3  X  —  {2x  —  x  +  a)} 
—  4,x  —  {3  X  —  2  X  -{-  X  —  a} 
=  4:X  —  3x+2x  —  x  +  a 
=  2x  +  a 
In  the  above  process  the  vinculum  was  removed  first,  and  then  the  brackets 
in  succession,  beginning  with  the  inner  one. 

If  we  remove  the  outer  bracket  first,  the  work  will  appear  as  follows ; 
4  X  -  3  .r  +  (2  ar  -  a:  -  a)  -  (l)  ' 

=  4x  —  3x  +  2x-x  +  a    .  .(2)  „ 

As  in  ( 1 )  the  +  sign  appears  before  the  bracket,  this  might  have  been  omitted 
at  once  without  change  of  signs,  and  at  the  same  time  the  i-inculum  over 
a:  —  a  omitted,  and  the  final  expression  obtained  at  once.  Thus  from  (1)  we 
have  at  once  4x  — Sx  +  2x  —  x  +  a  =  2x  +  n 

While  tlie  beginner  may  secure  accuracy  by  the  longer  method,  it  is  recom- 
mended that  the  more  advanced  student  begin  with  the  outermost  bracket, 
as  the  shorter  method. 


58  ALGEBRA. 

Remove  the  brackets  and  reduce  each  of  the  following  expres- 
sions to  its  simplest  form  : 

1.  a-\-  b  +  (3a-2b).  2.    a -]- b  -  {a  -  3b). 

3.  4a-  {3  a-  (2  a -a)}. 

4.  x+  I2x-  {3x+  {4x-6 a-)}]. 

5.  5-[4+{3-(2-x)}]. 

6.  a  —  {a  —  (a  —  a  —  b)} . 

7.  7  a  -[ob-  {4  a-  (3  a -2  Z»)}]. 

8.  a  -  (b  -c)-  \b-(a  -c)  -  [«  -  {2b-{a-  c)}]  j. 

55.  Conversely,  from  §  52,  we  have  for  the  introduction  of 
brackets  the  following 

Kiile. 

When  the  bracket  introduced  is  preceded  hy  the  plus  sign, 
all  the  terms  enclosed  must  he  written  without  chanrje  of  sign ; 
hut  ivhen  the  bracket  is  preceded  hy  a  minus  sign,  all  the 
terms  enclosed  must  he  ivritten  with  change  of  sign. 

According  to  this  rule  a  polynomial  can  be  written  in  a 
variety  of  ways. 

Thus,  a  +  b  -\-  c  —  d  +  f 

=  a-^{b  +  c-d+  f) 

=  a-  (-b-c  +  d-f) 

=  a  +  b^c-(d-f) 

=  a  +  b  —  {-c-i-  d—f) 

=  etc. 

Place  in  brackets,  with  the  sign  —  prefixed,  without  chang- 
ing the  value  of  the  expression  : 

1.  The  last  two  terms  of  a  +  b  —  c  -\-  d. 

2.  The  last  three  terms  of  a  —  b  —  c  —  d. 

3.  The  first  three  and  the  last  three  terms  of  —  a  —  b  -[-  c  — 
d  +  e-f 

4.  The  last  four  terms  of    —  a  —  Ji  —  c  -{-  d  -[-  p. 


MULTIPLICATION.  59 

CHAPTER    VIII. 
MULTIPLICATION. 

56.  Multiplication  is  a  short  method  of  finding  the  sum 
of  tlie  repetitions  of  a  number. 

57.  A  Factor  is  any  one  of  several  numbers  which  are  to 
be  multiplied  together  to  form  a  'product. 

58.  Any  one  or  more  of  the  factors  which  go  to  make  up 
the  product  may  be  called  the  Coefficient  of  the  remaining  fac- 
tors. Thus,  in  5  abc,  5  is  the  coefficient  of  a  h  c,  or  h  c  the 
coefficient  of  5  a,  or  5  a  &  the  coefficient  of  c,  and  so  on. 

A  coefficient  is  -called  numerical,  literal,  or  mixed,  accord- 
ing as  it  is  a  numeral,  a  letter  or  letters,  or  a  numeral  and 
letters  combined.  The  three  cases  above,  taken  in  their 
order,  illustrate  this. 

By  coefficient,  the  numerical  coefficient,  together  with  the 
sign  of  the  expression,  is  usually  meant.  If  no  figure  is 
expressed,  a  unit  is  understood  ;  thus,  x  is  the  same  as  1  x. 

59.  The  Reciprocal  of  a  number  is  a  unit  divided  by  that 
niimber  ;  thus,  the  reciprocal  of  3  is  |- ;  of  x,  -  . 

60.  A  Power  is  the  product  obtained  by  repeating  a  num- 
ber a  given  number  of  times. 

61.  An  Index,  or  Exponent,  is  a  number  either  positive  or 
negative,  integral  or  fractional,  placed  at  the  right,  and  a 
little  above  the  number. 

If  the  index  is  positive  and  integral,  it  indicates  how  many 
times  the  number  enters  as  a  factor  into  the  power. 


60  ALGEBRA. 

Thus,     2«  =  2x2x2x2  =  16;  read,  2  fourth  j^ower,  or  the 

fourtli  power  of  2. 

0,2  —  a  X  a ;  read,  a  second  power,  or  a  square. 

a^  =.  a  Y.  a  y.  a\  read,  a  third  power,  or  a  cube. 

a"  =  a  X  a  X  a  ...  to  n  factors;  read,  a  nth  power. 

Exponents  and  coefficients  must  be  carefully  distinguished. 

Thus,  x*  =  xxxxxxx 

while  4:X  —  X  +  X  +  X  +  X 

62.  ab  =  ha;  and  ahc  =  ach  =  hca;  or  the  product  of 
any  number  of  factors  is  independent  of  their  order. 

63.  Moreover  a  {he)  =  {ah)  c  =  b  (c a) ;  that  is,  the  product 
of  any  number  of  factors  is  independent  of  the  order  of  group- 
ing them. 

64.  If  d  is  a  positive  integer,  then  {a  +  b  +  c)d  = 

(a-\-  b  +  r)  +  (a  -^  b  +  c)-]-  {a-]-  h  +  c)  .  .  . 
repeated  d  times, 

=  a  +  a  -\-  a  -\-  .  .  .  repeated  d  times, 
+  b^b  +  b-\-  ... 

+  C  +  C   +  C+...  "  " 

:=ad  +  bd-\-cd; 

that  is,  the  product  of  the  sum  of  any  number  of  algebraic 
numbers  by  a  third  is  equal  to  the  sum  of  the  products 
obtained  by  multiplying  the  numbers  separately  by  the  third. 

65.  The  multiplier  must  always  be  an  abstract  number, 
and  the  product  is  always  of  the  same  nature  as  the  multi- 
plicand. 


MULTIPLICATION.  61 

The  cost  of  7  pounds  of  tea  at  60  cents  a  pound  is  60  cents 
taken,  not  7  pounds  times,  but  7  times ;  and  the  product  is  of 
the  same  denomination  as  the  multiplicand  60,  viz.,  cents. 

In  Algebra  the  sign  of  the  multiplier  shows  whether  the 
repetitions  are  to  be  added  or  subtracted. 

1.  (+  a)  X  (+4)  =  +  4a; 
that   is,    +  «  added   4  times   is    -\-  a  +  a  +  a  +  a  =  -^  A  a. 

2.  (+a)  X  (-4)  =  -4«; 
that  is,  +  (i  subtracted  4  times  is  —  a  —  a  —  a  —  a  =  —  i  a. 

3.  (_rt)  X  (+ 4)  =-4a; 
that    is,    —  a    added   4   times    is   —  «  —  «  —  a  —  «  =  —  4  a. 

4.  (_f,)  X  (- 4)  =  +  4a; 
that  is,  —  a  subtracted  4  times  is  +  «  +  «  +  «  +  «  =  +  4  a. 

In  tlie  first  and  second  examples  the  nature  of  the  product 
is  +  ;  in  the  first,  tlie  +  sign  of  4  shows  that  the  product  is 
to  be  added,  and  +  4  a  added  is  +  4 « ;  in  the  second,  the 

—  sign  of  4  shows  that  the  product  is  to  be  subtracted,  and 
+  4  «  subtracted  is  —  4  a.  In  the  third  and  fourth  examples 
the  nature  of  the  product  is  — ;  in  the  third,  the  +  sign  of 
4  shows  that  the  product  is  to  be  added,  and  —  4  a  added  is 

—  4  a  ;  in  the  fourth,  the  —  sign  of  4  shows  that  the  product 
is  t(j  be  subtracted,  and  —  4  a  subtracted  is  +  4  «. 

66.  Hence,  in  multiplication,  we  have  for  the  sign  of  the 
])roduct  the  following 


Like  sifjns  give  +  ;  unlike,  — . 

Hence  the  product  of  an  even  number  of  negative  factors 
is  positive  ;  of  an  odd  number,  negative. 


62  ALGEBRA. 


MULTIPLICATION    OF   ALGEBRAIC    LITERAL 
EXPRESSIONS. 

Case    I. 
67.     When  the  Factors  are  Monomials. 

1.  Multiply  2  a  by  o  b. 

2ax5b  =  2xaXi)Xb  —  2x5X(iXb  =  10ab 
As  the  product  is  the  same  in  whatever  order  the  factors  are  arranged, 
we  have  simply  changed  their  order  and  united  in  one  product  the 
numerical  coefficients. 

2.  Multiply  a^  hy  a\ 

As  the  exponent,  if  integral  and  positive,  of  a  number  shows 
how  many  times  it  is  taken  as  a  factor, 
«3  =  a  X  a  X  a 
and  a-  =  a  X  « 

.  • .    a^  X  d^—  {a  X  a  X  a)  X  (a  X  a)  =  a  X  a  X  a  X  a  X  a  =  a^ 

Therefore  the  product  of  powers  of  the  same  number  is  f/iat 
number  with  an  index  equal  to  the  sum  of  the  indices  of  the 
factors.     This  is  called  the  Index  Law. 

Hence,  when  the  factors  are  monomials,  we  have  the  following 

Rule. 

Annex  the  product  of  the  literal  factors  to  the  product  of  their 
coefficients,  rememherinfi  that  like  signs  give  +,  and  unlike,  — . 

(3.)  (4.)  (5.)  (6.)  (7.) 


5x              Ban 

2y              4ab^ 

10 xy         20  a"  b^ 

C^x 
-    3x' 
-18a-=^ 

if 

-2\yz^               -2x'  y" 

^hy''               -Ix^y- 

-^^hy^  z"              14  a:^  ?/« 

8.    a'  Xa'  =  ? 

12. 

- 

a-}  X  (-  a«)  =  ? 

9.    X^  X  .T^  =  ? 

13. 

4  a'"  X  2a"  =  ? 

10.    c3  X  (-  r-2)  =  ? 

14. 

a!" 

X  a"  =  ? 

11.    -x^  X  x^  =  ? 

15. 

10 

_^2  ^8  X  4  rr^  yA  ^  9 

MULTIPLICATION.  63 

i6.~7aH^Xoa^b''=?  19.  —  12 ,/;' s-  X  (—  3 x'  jf  z)  =? 

n.-ix'fx(-3 x^  f)  =  ?      20.  40  a^ b^X{-b a* b°)  =  ? 
18.   12a;^xyx^a^x'ij-^-!       21.  ^ d'b^c'x^aHc^  =  '^ 

22.  babx^bcX'So  b^'i 

23.  6  a-  c"  cZ^  X  3  i^  c*  X  (-  5  a*  b')  =  ? 

24.  7  a-^  ^  2-  x(-2x'  if  z')x{-'6  X*  if  z")  ^  ? 

25.  2  (x  +  y)  X  6  (a;  +  y)-  =  V 

Case    II. 

68.     When    only   one    Factor   is   a    Monomial. 

1.    Multiply  a  +  b  -\-  c  by  x. 

{a  +  b  -\-  <■)  x  =  ax  -\-  b  X  -\-  ex 

or,  a  -{-  b  +  c 

X  These  results  follow  from  §  64. 


ax  -{-  b  X  +  c  X 

Therefore,  for   the   multiplication   of  a  polynomial  by  a 
monomial,  we  have  the  following 

Rule. 

Multiply  each  term  of  the  ^mtltijjlicand  hy  the  multiplier, 
and  connect  the  several  results  by  their  proper  signs, 

(2.) 
'Sx'  ■\-  ox  —  1  y 
ryxy 


15  x^  y  +  25  X"  y  —  35  x  y"^ 

3.  (a^  +  3  a-  +  4  a)  X  2a=  ? 

4.  («2  -2ab  +  b-)  X  ab=\' 

5.  (3  cc^  -  5  X-  +  2  ./•)  X  5  x'  =  ? 


64  ALGEBRA. 

6.  {a"  -  b^  -  C-)  X  "bc  =  ? 

7.  (2x^  —  ox^  —  x)  X  {—5x^)=? 

8.  (5  x-  (/  —  6  X  //'-  +  8  X-  I/-)  X  '^x  1/  =  ? 

9.  (a^  —  3  a^  b-  -^  U^)  X  '6  a-  ^  =  ? 

10.  (8  *5  ^1°  -  7  a;3  //«  +  4  cc-  //*  -  3  x'  if)  x  (-2xf)=? 

11.  (a.-'^  —  3  X'-  ?/  +  3  X-  ^-  —  /)  X  X'  1/  =  ? 

12.  {—  b  X  (J-  z  -{-  '6  X  y  si'  —  '6  X-  y  z)  x  xy  z  =  'i 

Case    III. 
69.     When   the    Factors   are    Polynomials. 

1.  Multiply  a  -\-  b  -\-  c  by  x  -\-  y. 

{a  +  b  +  c)  {x  +  y)^a{x-\-y)  +  b  {x  +  y)  +  r  {x  ^  y) 

(x  +  y)  is  here  regarded  as  a  single  term.  The  last  expression  further 
reduced  becomes  ax  +  ay  +  b  x  +  h y  +  c x  +  c  y,  and  this  equals 
a  X  +  b  X  +  c  X  -\-  a  y  +  b  y  +  c  y. 

Hence,  for  the  multiplicatiou  of  a  polynomial  by  a  polyno- 
mial, we  have  the  following 

Kule. 

}fidtiply  each  term  of  the  riiultiplicand  hy  each  term  of  the 
multiiilicr,  and  find  the  sum  of  the  several  jJrodiicts. 

2.  Multiply  3  a^  -  2  «  i  +  4  6-  by  2  a  -  3  b. 

3a-—    2ub   +     4:b-  We   begin   at   the   left, 

2a—    o  b  placing  the  second  result 

a    3         A    2~7    I      c     P  one  place  to  the  right,  so 

_    „  -  ^      ,  o       ^  o  7  -  that  like  terms  may  stand 

—    9  a-b  +     bab- —  iJ  b'  .      ,,  ,.    ,       , 

m  the  same  vertical  col- 


6  r/S  -  13  a^  6  -f  14  a  6^  -  12  b^ 


uran. 


MULTIPLICATION.  65 

3.    Multiply   2  a;  —  5  a;"-"  \-  2  -\-  x^  by   2  x''  +  2  —  x. 

In  Examples  2  and  3  the 

x^ —    5x^+    2x  +    2  multiplicand    and    multi- 

2x^ —       x  +    2  plier  were  arranged  accord- 

2a;»  — 10a;*+    4a;^+    4a--  "^S  ^^  ^^^  descending  pow- 

—      a;*+    5x^—    2aj^  — 2  a;  ers  of  a;  before  multiplying. 

„    ,       wJ!    o   ,    T      ,    ,  The     polynomials     could 

2a;2  — 10a;^  +  4.T  +  4  u         \                      i 

—  have    been    arranged    ac- 

wa;'      11a;  +  11a;  5a;-  +  ^a;4-4         cording  to  the  ascending 

powers  as  well. 

4.  (a;  +  a)   X  (a;  +  />)=? 

5.  (2a  +  b)  X  {2a-\-  3b)=? 

6.  (3  a-  —  4  «  +  5)  X  (2  a  +  5)  =  ? 

7.  (cr  -{-  ab  +  b")  X  (cr  —  ab  +  b")  =? 

8.  (x^-x-l)x(x-l)  =  ?  Ans.    a;3-2a;2  +  l. 

9.  (a-  —  2  a  a;  +  4  a;'^)  X  (fr  +  2  a  a;  +  4  a;"^)  =  ? 

10.  («2:,;  _  «a;'^  +  a;3  -  a^)  x  (a.-  +  a)  =  ?         Ans.    a;"  -  a*. 

11.  (16  a'  -{-  12  a  b  +  d  b')  X  {4.  a  -  3  b)  =  ? 

12.  (a;'^  +  a;  -  2)  X  (a;''  +  a;  -  (5)  =  ? 

Ans.     x^  +  2x^-7x^-8x^12. 

13.  (2  .*;3  -  3  x^  +  2x)  X  (2  a;-'  +  3  a;  +  2)  =  ? 

14.  {x^  —  3  a;-  +  3  a;  —  1)  X  (a;-  +  3  a;  +  1)  =  ? 

Ans.     x^  -  5  x^  -{-  5  x^  -  1. 

15.  {-a^  +  a*b  -a""  b')  X  {-  a  -  b)  =  ? 

16.  (r(3  +  2  a^  ^»  +  2  a  b')  X  {a-  -  2  a  b  +  2  b")  =  ? 

Ans.      a.^  4-  4  a  b*. 

17.  {x^-3xy-  y-)  x  {- x' -{■  x  >j  +  y^)  =? 

18.  (6^  -  a'  b'  +  a**)  X  («^  +  a'  6'  +  b^)  ='? 

Ans.     a«  —  a^  5*  +  2  «»  6^  +  b^. 

19.  (a;2  -  2  a;  y  +  //)  X  (x^  +  2  a;  y  +  ?/-)  =  ? 

20.  (««  -  3  a'^  6  +  3  «  i'^  -  ^<3)  X  {a  +  b)  =? 

Ans.     a^-2a3^  + 2a$3_54_ 


66  ALGEBRA. 


CHAPTER    IX. 
DIVISION. 

70.  Division  is  finding  a  quotient  which,  multiplied  by 
the  divisor,  will  produce  the  dividend.  Division  is  the 
inverse  of  multiplication. 

In  accordance  with  this  definition  and  the  Eule  in  §  66, 
the  sign  of  the  quotient  must  be  +  when  the  divisor  and  the 
dividend  have  like  signs;  and  —  when  the  divisor  and  the 
dividend  have  unlike  signs ;  that  is,  in  division,  as  in  multi- 
plication, we  have  for  the  signs  the  following 

Rule. 

Like  signs  give  +;  unlike,  — . 


DIVISION  OF  ALGEBRAIC  LITERAL  EXPRESSIONS. 

Case  I. 
71.     When   the    Divisor   and    Dividend    are    both    Monomials. 

1.  Divide  8  ax  by   2  x. 

The  coefficient  of  the  quotient  must  bo 

8«a;-:--a;  =  4a  a  number   which,   multipHed   by   2,   the 

coefficient  of  the  divisor,  will  give  8,  the 

coefficient  of  the  dividend,  that  is,  4;  and  the  literal  part  of  the  quotient 

must  be  a  number  which,  multiplied  by  ar,  will  give  a  x,  that  is,  a ; 

the  quotient  required,  therefore,  is  4  a. 

2.  Divide  a"  by  a\ 

aa  aaaa  a  . 

a'  -^  a^  =^  ar^   or  =:  aa  a  a  —  «* 


DIVISION.  67 

For  (§  67).  a*  X  a^  —  aaaa  X  aaa  =  a'.  Therefore  the 
quotient  of  two  powers  of  the  same  number  is  that  number  with 
ail  index  equal  to  the  index  of  the  dividend  minus  the  index  of 
the  divisor. 

Hence,  for  the  division  of  monomials  we  have  the  following 

Rule. 

Annex  the  quotient  of  the  letters  to  the  quotient  of  their  cocf- 
Jlcicnts,  rememherinrj  that  like  signs  give  +  and  unlike,  — . 

(3)  (4) 

14a^c-^  =  2  a-^  6^  c  ^  ^'  ^*  ^'  -  -^  ^-  "'  -^ 

7  abc 

(5) 
—  15  a^  x^  r. 


Zxyz"-             -" 

(6) 

—  '  a  c 

-    5a' e' 

14. 

^5  ,n  _^  ^2  Hi  _  V     Ans.  a^ 

15. 

a'"  --  a-  =  ? 

16. 

«'"  ^  a"  =  ? 

3a^  X 

7.  aH'^c-^  (a-  b-)  =  ? 

8.  d^ b  c  -=r  (—  abc)  =^  '.' 

9.  —  a^x*  -^  (—  d^x)  =  ■; 

10.  3oa''bc''-^{-5ac')=?       17.   a'"  ^  a  =  ? 

11.  U¥x'u-^{-2xii)^?       18.    fl''+'-^a»  =  ? 

12.  4  a'  b'  c"  ~  {ab'c')  =  ?         19.    a^'+^  -f-  a"  =  ? 

13.  3  a  ^  3  =  ?  20.    6  a'"  6"  -^  (3  a  b'-)  =  ? 

21.  (a  +  6)^  ^  (a  +  by  =  ?  Ans.    (a  +  by. 

22.  9(a+  6)5-^  {:^{n-j-b)'}  =  ? 

23.  -  12  (X  +  i/Y  -  {4(^-  +  yr}  =  ? 

24.  -  10  (x  +  >/r  ^  {-  2  {x  +  i/Y"}  =  ? 

25.  18  (x  -  zy  ^  {-6(x-  zY}  =  ? 

26.  27  (a  +  2)«  -  {9  (a  +  2)}  =  ? 

27.  33  (1  -  xy  -^  {-  11  (l--.ry}  =  ? 


68  ALGEBRA. 

Case    II. 
72.     When   the   Divisor   only    is   a    Monomial. 

1.    Divide    ax-{-at/-\-az    by    a. 

From  §  68,  {a  x  +  a  //  +  a  z)  -^  a  =  x  +  y  -\-  z.  That  is, 
the  quotient  obtained  by  dividing  the  sum  of  two  or  more  mono- 
mials by  another  is  the  sum  of  the  quotients  obtained  by  dividing 
the  monomials  separately  by  this  other.     Hence,  the  following 


Divide  each  term  of  the  divideiul  hj  the  divisor,  and  connect 
the  several  results  hy  their  proper  signs. 


(^•) 

(3.) 

3a')15aH^-9aH* 

-4.x)- 

■12x^1/^-8x^1/ 

5P-3ab* 

?,  x^  if -^  1  x"  if 

4.     {a^  +  ab)  ^a  =  ? 

5.    (6a3 

-3aH)  -^  (3«)  =  ? 

6.    (x'-7x'  +  A  x')  -^ 

x^^ 

=  ? 

7     (15ic5-25.x^)  -^  (- 

-5x 

:^)=? 

8.  (-  24  a:«  -  32 .7;*)  -^  (-  8  x^)  =  ? 

9.  (a"  +  a^b  +  a-  P)  H-  cr  =  ? 

10.  («"  -  aH -{-  a^  b^)  ^  a^  =  ? 

11.  (x'  -x*  +  x''-  a;2)  -^  a;2  =  ? 

12.  (a- h  c  ■}-  ab'^c-^  a b  c")  -f-  (o  b  r)  =  ? 

13.  (-  a^z^  -  ^2-2  _  ,,  ^)  ^  (_  „  ,,)  ^  9 

14.  (-  34  x«  +  •'^l  x^  -  17  «  a;)  ^  (17  a;)  =  ? 

15.  (6a;V--8x«/  +  12a;2  3/4)  ^  (Oa;?/)  =  ? 

16.  (6  a;"  //  -  8  r«  //  +  12  x""  if)  ^  (-  2  a:^  y^)  :^  ? 

17.  (&2  a;2  ,^2  _  ^,3  x^  ^^  +  b"  x^  s")  ^  (-  b""  a;2)  =  ? 


DIVISION.  69 

Case    III. 
73.     When   the    Divisor   and    Dividend   are   both    Polynomials. 

1.    Divide  (i^  —  o(i'^  b  ^  o  a  li'  —  1/  by  a'  —  'lah-\-  b'-^. 

a?  —  2  a  b  -\-  b'^)a^  —  o  a'-  b  +  3  a  />'-  —  b^  («  —  b 
aS  —  2(rb-\-     ab' 

—  (r  b  -\-  -  a  I)'-  —  b"^ 

-  d'b  +  '2  a  b'  -  b^ 

The  divisor  and  dividend  are  arranged  in  the  order  of  the  powers  of  a, 
beginning  with  the  highest  power.  a%  the  highest  power  of  a  in  the 
dividend,  must  be  the  product  of  the  liighest  power  of  a  in  the  t^uotient 

and  a^  in  the  divisor ;  therefore       =  «  must  be  the  highest  power  of  a 

«■- 

in  the  quotient.  The  divisor,  o- —  2  «  6  +  6'^,  multiplied  by  a,  must 
give  several  of  the  partial  products  which  would  be  produced  were  the 
divisor  multiplied  by  the  whole  quotient.  When  (a^  —  2  ab  -\-  b^)  a 
=  a^  —  2a^b  +  a  b-  is  subtracted  from  the  dividend,  the  remainder  must 
be  the  product  of  the  divisor  and  the  remaining  terms  of  the  quotient ; 
therefore  we  treat  the  remainder  as  a  new  dividend,  and  so  continue 
until  the  dividend  is  exhausted. 

Hence,  for  the  division  of  polynomials,  we  have  the  following 


Arrange  the  divisor  and  dividend  in  the  order  of  the 
powers  of  one  of  the  letters. 

Divide  the  first  term  of  the  dividend  hy  the  first  term 
of  the  divisor;  the  result  will  be  the  first  term  of  the 
quotient. 

Multiply  the  whole  divisor  hy  this  quotient,  and  subtract 
the  product  from  the  dividend. 

Consider  the  remainder  as  a  new  dividend,  and  proceed  as 
before  until  the  dividend  is  exhausted. 


70  ALGEBRA. 

2.  Divide  a;-  +  H  «  +  30  by  x  +  5. 

X  +  o)x'  +  11  x  +  30  (x  +  6 
x'  +  5x 

6  a;  +  30 
6  a; +  30 

3.  Divide  8  a^  +  8  a'-^  6  +  4  a  h-  +  P  hy  2a-\-  b. 

8  ./.^  +  8  a^  6  +  4  a  ^»--^  +  ^»M  2  a  +  b 


8  a-^  +  4  g-^  b  I  4  a2  +  2  a  6  +  62 


4 

4 

a-  ^»  +  4 

aH  +  2 

2 
2 

ab'-  +  b^ 
ab^  +  b' 

4.    Divide  12  a^  +  10  o-  -  4  +  8  a  -  26  a^  by  2  a-  +  1  -  3  a. 

Arrange  according  to  the  descending  powers  of  a. 

12  a^  -  26  a^  +  10  a^  +  8  a  -  4    2  a^  -  3  a  +  1 
12  a^- 18  0^4-    6a2  |6a-^-4a-4 


-  8a^ 

-  Sa^ 

+    4  a^  +    8  a 
+  12  a^  _    4  a 

— 

8  a^  +  12  a  - 
8  a^  +  12  a  - 

-4 
-4 

5.    Divide  a*  —  b*  by  «  —  b. 

a-  b)a'-b'{a^  +  a-  b  -\-  a  b^  -\-  b^ 
a4  -aH 


aH>- 

an- 

-b* 
-d'b-' 

a-  //-  - 
aH'- 

-b* 
-ab"" 

ab""- 
ab^- 

-b* 
-b' 

DIVISION. 

6.    Divide  x'"  -  3  x"  f  +  2  f "  by  x»  -  2f. 

x'^-y'^)x'"-3x'^^J"  +  '2r"{x•'- 

-2r 

^ln_        ^nyn 

-2x"ij" +  2>j-" 

-2x»y«  +  22/-^» 

71 


7.    Divide  a""  +  b'  +  e^  -  3abc  by  o  +  ^v  +  t". 


o  a 


hr  -{-  b""  +  c^\a+  b  +  c 


a?  b   +  a^e        \  d^  —  a  b  —  a  r  -^  b-  —  b  c  +  & 


—  cv-b  —  a-  c  —  3ab<' 

—  a-b   —  a  b'^  —      «  b c 


—  d'  e  +  a  b'-  — 

—  d^c              — 

'labc 
a  be  —  a  c^ 

ab' 

abr  +  or--+  b^ 

+  U^  +  b-'c 

I 

abr  +  a  (r  —  b'^  c 

abc              -b-'c-bc- 

ae'  +  bf'^  +  e^ 
a  C-2  +  b  r^  +  c^ 

In  the  examples  the  dividend,  divisor,  and  successive  remainders  are 
arranged  in  descending  powers  of  a  and  x.  The  ascending  powers  would 
have  answered  as  well.  The  choice  of  letter  and  kind  of  arrangement 
are  immaterial,  but  it  is  especially  important  before  beginning  the 
division  that  some  arrangement  shoiild  be  adopted  and  maintained 
throughout  the  operation. 

8.  (x^  -11  x  +  72)  ^  (a;  -  9)  =  ?  Ans.     a:  -  8. 

9.  (9a;'-  -  3x  -  2)  -^  (3  a:  -  2)  =  ? 

10.  (6a-2  -  5  a;  -  6)  -^  (2a;  -  3)  =  ?  Ans.     3.^  +  2. 

11.  (9.r3  -  18a--^  +  26a;  -  24)  -^  (3a;  -  4)  =  ? 

Ans.     3a'2-2a;  +  6. 

12.  (.t3  +  27)  -^  (a;  +  3)  =  ? 

13.  {x^  -  27)  ^  {0^  +  3  a;  +  9)  =  ?  Ans.     a;  -  3. 


72  ALGEBRA. 

14.  (x3_^3)^(^_^)^9 

15.  {x'  +  /)  ^  (X  +  y)  =  ? 

16.  (a;«  -  /)  --  (x^  +  x-^  f  +  y')  =  ? 

17.  (x^  +  a;"  +  1)  ^  (a;-^  +  a;  +  1)  =  ? 

18.  (x'«  -  y«)  -  (x-^  +  xy  +  y')  =  ? 

Ans.     X*  —  x^  y  -\-  X  y^  —  ,?/. 

19.  {x^  -x'y  +  xf  -  /)  -^  (X  -  2/)  =  ? 

20.  (G  x*  +  23  x-3  +  42  x-  +  41  a-  +  20)  ^  (3  x^  +  4  a;  +  5)  =  ? 

Ans.     2  a--  +  5  a;  +  4. 

21.  (6  a;*  +  32  x'^  -  23  a;=^  -  18  -  9  a;)  -^  (6  -  5  a;  +  2  x^)  =  ? 

22.  (72a:-^+  20x^-  33^=^-353:  +  30)  -^  (la;^- 5a;  +  10)  =  ? 

Ans.     5  a;-  —  2  a;  +  3. 

23.  {2y'  _  16  3,3  +  2  /  +  92  3/  +  48)  -  (-  5  y  -  12  +  /)  =  ? 

24.  (80  a  _  23  «3  +  3  a^  -  5  «'^  +  50)  ^  («-  -  6  «  -  10)  =  ? 

Ans.     3  (1^  —  Ij  a  —  b. 

25.  (x" - 8 x^y  +  21  x"" y- - 16 X  ?/ -  7 ^ )  -^  {x^ -bxy-\-7y^)  =  ? 

26.  (x*-9oxH12«^T2  +  35a^x  +  15a^)-^(a,••■^-4aI«-3a-)  =  ? 

Ans.     X-  —  bax  —  5a^. 

27.  (40aZ-3_^25//-4«^//-^+4a^-16«3/y)^(2o-^-4a&-562)=z? 

28.  (4y^-15/>3,3+  26/>V-23//y+  8Z/*)^(4?/--7//?/  +  8^--0=? 

Ans.     y"  —  2  by  ^  h'-. 

29.  (31  x^  5/2  +  5  .^4  +  12  y^  -  14  x^  y-22x  /) 

^(_4xy  +  5a:2+ 3^/2)  =? 

30.  (6  a*  +  21  a=^ h  +  31  rf ^ ^;2^_  27  « /,3_ 5 ^4)^(3 ^2_^  g „  j _ j2) ^o 

Ans.     2(r  ^3ab  +  b I/\ 

31.  (a^  +  ^'^  +  2  a%^  _  2  r>2^2  _  ^4  _  ,/4)  ^  (,,2  ^  ^y2  _  ,.2  _.  ,/2)  ^  9 

32.  (6a:*  +  5a;''  +  6a;2-17x4-6)  -^  (6aj2-7a-  +  2)  =  ? 

33.  (9  a'  h^  -  12  a"  h  +  3  /.^  +  2  o^  52  _^  4  „5  _  n  ^,  ^,4^ 

^  (.3  ^3  _^  4  ^,3  _  2  a  ^»2)  ^  ? 

34.  (a«  -  3  a"  b-'  +  3  (r'  b'  -  Z»«)  -f-  (rr'^  +  3  a^  b  +  ^i  a  P  +  P)  =  ? 


THEOREMS  OF  DEVELOPMENT.  73 


CHAPTER    X. 
THEOREMS    OF    DEVELOPMENT. 

74.  Let  a  and  b  represent  any  two  numbers.  Their  sum 
is  a  +  &  ;  and  {a  +  b)^  =  (a  +  b)  (a  +  b),  which  expanded  be- 
comes a^  +  2ab  -{-  b^,  as  appears  from  the  following  process  : 


a   +  b 

a  +  b 

a^  +     ab 

+      ab  +  b'^ 

a^ -\- 2  a  b -\- b'^ 

this 

we 

dedi 

ice  the  following 

THEOREM. 

TJie  square  of  the  sum  of  two  numbers  is  equal  to  the  square 
of  the  first,  plus  tivice  the  jJroduct  of  the  two,  plus  the  square 
of  the  second. 

According  to  this  theorem  find  the  square  of 

1.  x-^y.  8.  ^x  +  5y. 

2.  X  +  1.  9.  2  X  +  3  y. 

3.  X  +  2.  10.  x^  +  y\ 

4.  3  x  +  1.  11.  3  x^  +  2/- 

5.  a  cc  +  1.  12.  2  a  +  5  b. 

6.  2a;  +  a.  13.  3.t  +  4?/. 

7.  a2  +  6^  14.  5a; +  ^. 


74  ALGEBRA. 

75.  Let  a  and  h  represent  any  two  numbers.  Their  differ- 
ence is  a  —  h\  and  {a  —  bf  ^  («  —  b)  (a  —  b),  which  ex- 
panded becomes  a'^—2ab  +  b^,  as  appears  from  the  following 
process : 


a 

-h 

a 

-b 

a' 



ah 

- 

ab  +  h'' 

a^  -2ab  +  b'' 
From  this  we  deduce  the  following 

THEOREM. 

77*6  sqyiare  of  the  difference  of  tivo  numbers  is  equal  to  the 
square  of  the  first,  minus  twice  the  2)'>'oduct  of  the  two,  j^ius 
the  square  of  the  second. 

to  this  theorem  find  the  sqviare  of 


According  to  tl 

1. 

X  —  a. 

2. 

x-1. 

3. 

x-2. 

4. 

2x-a. 

5. 

o2  _  b\ 

6. 

2x-^y. 

7. 

4x-3y. 

8. 

x'-f. 

9. 

ab  —  xy. 

10. 

2x^-y. 

11. 

5z-3y. 

12. 

3  a  -  5  X. 

13. 

5x-2z. 

14. 

1  z  —  y. 

15. 

3ab-2c. 

16. 

2  a  c  —  b. 

17. 

4x-4y. 

18. 

Sz-2y. 

19. 

5fi-5b. 

20. 

4  ic  —  6  y. 

21. 

8a -7  b. 

22. 

2z-8y. 

23. 

x-9y. 

24. 

2c -5d. 

25. 

5a -4  b. 

26. 

3  a^  -  bK 

27. 

4an^-bc. 

28. 

6x'-5y\ 

29. 

7a^-3  b\ 

30. 

9  x^  -  z"". 

THEOREMS   OF   DEVELOPMENT.  75 

76.  Let  a  and  h  represent  any  two  numbers.  Their  sum 
is  a  +  h,  and  their  difference  a  —  h\  and  {a  +  b){a  —  b) 
=  «2  _  52^  as  appears  from  the  following  process  : 


a-  -  b' 

From  this  we  deduce  the  following 

THEOREM. 

The  product  of  the  sum  and  difference  of  two  numbers  is 
equal  to  the  difference  of  their  squares. 

According  to  this  theorem  multiply 

1.  X  -\-  y   hj   X  —  y.  6.    2  a  +  i   by   2  a  —  6. 

2.  a;  +  3   by   X  -  3.  7.    3  +  ic   by   3  -  cc. 

3.  «  +  1   by   «  —  1.  8.    5  +  y  by   5  —  //. 

4.  x^  +  3/2   by   x^-y''.  9.    2  ?/  +  7   by   2  ^  -  7. 

5.  2  a  +  3  6  by  2  «  -  3  J.        10.    6  «  +  6   by    6  a  -  6. 

77.    This  theorem  suggests  an  easy  method  of  squaring 
numbers. 

For,  since  a'^  =  {a -\-  b)  (a  —  &)  +  b'^, 

49-  =  (49  +  1)  (49  -  1)  +  12  =  50  X  48  +  1 

=  100  X  24  +  1  =  2401 
In  accordance  with  this  principle  find  the  square  of 

1.  98. 

982  ^  ^98  +  2)  (98  -  2)  +  2^  =  100  x  96  +  4  =  9604 

2.  99.  4.    48.  6.    499.  8.    999. 

3.  98.  5.    47.  7.    493.  9.    991. 


76  ALGEBRA. 

78.  Let  X  +  a,  x  +  b  represent  any  two  binomials.  Then 
(x  -\r  a)  {x  +  b)  =  a^  -\-  (a  -{-  b)  x  +  a  b,  as  appears  from  the 
following  process : 

X  +  a 
X   -{■  b 

x^  +  ax 

+  b  X  +  a  b 


x^  +  {a  +  b)x  +  ab 
From  this  we  deduce  the  following 

THEOREM. 
The  product  of  two  binomials  of  the  form,  x  -\-  a,  x  +  b, 
is  equal  to  the  square  of  the  first  term,  plus  the  sum  of  the 
second  terms  into  the  first  term,  plus  the  product  of  the  second 
terms. 

(1-)  (2.) 


a;  +  5 

X  -5 

X  +3 

x  -3 

x^  +  5x 

x^-5x 

+  3a;  +  15 

-3a; +  15 

a:-2  +  8x  +  15 

a;2_8a:+  15 

(3.) 

(4.) 

X  -5 

X  +5 

X  -\-3 

a;  -3 

x^-5x 

x^  +  5x 

+  3a;-15 

-3a; -15 

a;2-2a;-15 

x'^  +  2  a;  -  15 

According  to  this  theorem 

multiply 

5. 

x  +  7   by    X  -\-  2. 

11. 

X  —  4  a   by    x  -|-  2  a. 

6. 

x-7    by   x-2. 

12. 

X  —  1   by   X  +  5. 

7. 

X  +  7   by    X  —  2. 

13. 

X  +  8   by   X  -  1. 

8. 

x-7   by   X  +  2. 

14. 

x  +  4a   by   x  +  2a. 

9. 

a  —  5   by    a  +  4. 

15. 

a  — 3  b    by    a -\-  5  b. 

LO. 

y  —  3x   by  y  —  5 X. 

16. 

X  +  6a    by    X  ~  iJa. 

THEOREMS  OF  DEVELOPMENT.  77 

79.    MISCELLANEOUS    EXAMPLES. 


Find  the  square  of 

1.    :r  +  2^. 

5.    2  a  +  a;. 

9.    2  a  X  +  3  //. 

2.    x-2y. 

6.    2a -x. 

10.    2ax-3yz. 

3.    o  +  3  c. 

7.   3  «•-  +  ^. 

11.    2  a;- +  32/-. 

4.    a-3e. 

8.    3«-^-2/. 

12.    2x''--^if. 

Find  the  product  of 

13.    X  -  10  and  x  +  10.               16. 

d^  4-  9cc-anda2-9c 

14.    X-  +  12  and  cc^ 

-  12.            17. 

2a-36cand2«  +  S 

15.    X  —  10  a  and  x 

+  10a.         18. 

a  -\-b  -\-  c  and  rt  +  6  - 

[a  +  i  +  c]  [«  +  ?j  -  f]  =  r(rt  +  5)  +  c]  [(<(  +  6)  -  ,•]  =  («  +  Z*)'-^  -  c^ 

(§'«)■ 

19.  a  +  Z»  4-  c  and  a  —  h  —  e. 

la  +  b  +  c]  [«  -  ?;  -  f]  =  [a  +  (6  +  0]  [o  -  (^  +  0]  =  ^ 

20.  a  +  b  —  c  and  o  —  />  +  '*• 
'21.    a  —  b  -{-  c  and  a  —  b  —  c 

22.  o  —  ^;  +  r  and  «  +  ^i  +  (•, 

23.  o-  +  a  +  1  and  «-  —  u  ■\- \. 

24.  «  +  2  ^»  +  3  f  and  «  +  2  ^»  —  3  c. 

25.  «  +  2  &  +  3  r  and  ri  -  2  />  +  3  c. 

26.  «  +  &  +  c  +  f^  and  a  ■\-  b  -\-  c  —  d. 

27.  cc"-^  +  »:?/  +  3/'-  and  x  —  //. 

28.  y?  —  ^  1/  -{-  y'  ii^iid  X  +  ?/. 

29.  x-  —  a,    X  +  o,  and  x-  +  «-. 

30.  a;  —  3,    x  +  3,  and  .r-  +  9. 

31.  2a;  —1,    2a-  +  1,  and  4a--  +1. 

32.  1  —  a  a?,    \  -\-  ax,  and  1  +  cr  a-'-. 

33.  a?  —  a,    a;  +  a,    a;'-  +  a^,  and  x^  +  «*. 

34.  a;  -  1,    a;  +  1,    .a--  +  1,    u"  +  1.  and  x-^  +  1. 


78  ALGEBRA. 


CHAPTER    XL 

FACTORING. 

(See  Preface.) 

80.  An  algebraic  expression  which  contains  no  terms  in 
tlie  fractional  form  is  called  an  integral  exjjression. 

Thus,  a^  —  b-,    oxy  ■\-  2z,  are  integral  expressions. 

81.  A  Root  is  one  of  the  equal  factors  into  which  a  number 
may  be  resolved. 

A  root  is  indicated  by  the  radical  sign  y,  the  initial  letter 
of  the  word  radix  (i-oof).  The  root  index  is  written  at  the 
top  of  the  sign,  though  the  index  denoting  the  second,  or 
square,  root  is  generally  omitted.     Thus, 

-y/a  ;  read,  the  second  root,  or  the  square  root,  of  a. 
's/a  ;   read,  the  third  root,  or  the  cube  root,  of  a. 
^/a  ;  read,  tlie  n\\\  root  of  a. 

An  expression  is  rational  when  none  of  its  terms  contain 
square  roots  or  other  roots. 

82.  Tlie  Factors  of  an  algebraic  expression  are  the  rational 
and  integral  expressions  whose  product  is  this  expression. 

83.  A  Prime  Factor  is  one  that  is  divisible  without  a 
remainder  by  no  rational  and  integral  expression  except 
i   itself  and   ±  1. 

84.  The  factors  of  a  purely  algebraic  monomial  are 
apparent. 

Thus,  the  factors  of  ac^x^  )jz  are  a.  c,  r,  x.  x,  y,  and  z. 


FACTORING.  79 

85.  Polynomials  are  factored  in  accordance  with  the 
principles  of  division  and  the  theorems  of  the  preceding 
chapter. 

Case  I. 

86.     When  all  the  Terms  have  a  Common  Factor. 

1.    Find  the  factors  of  b  x  -\-  b  ij  —  b  z. 

As  6  is  a  factor  of 
{hx  -^  hij  —  h'S)  =  h{x  -\-  y  ~  z)  each  term,  it  ninst  be 

a  factor  of  the  polyno- 
mial ;  and  if  we  divide  the  polynomial  by  h,  we  obtain  the  other  factor. 
Hence  the  following 


Divide  the  given  ijohjnomial  hy  the  common  factor ;  take 
the  qtcotient  thus  obtained  for  one  of  the  factors,  and.  the 
divisor  for  the  other. 

Note.  The  greatest  mouomial  factor  is  usually  sought.  The  two  factors 
may  often  be  still  further  resolved. 

Find  the  factors  of 

2.   x^  +  x'y  ->r  X  if.  14.  6  a^  +  2  a*  +  4  df>. 

Ans.     X,  and  x^-\-xi}-\if.  15.  2>  x'^  —  2>  x^  y  ■\-  ^  v?  f. 

16.  la -la?   +14  a\ 


3. 

a  X  +  a^. 

4. 

x^-Sax. 

5. 

x'  -  x\ 

6. 

2a-2a\ 

7. 

3  a2  +  6  a  6. 

8. 

15  +  25  x''. 

9. 

7  -  35  a;. 

10. 

13cc-393^. 

11. 

15  s  +  35  y. 

12. 

ah  —  abx. 

13. 

4  x2  -  2  X. 

17. 

5  o;^  -  10  a2  a;^  _  15  a^  x\ 

18. 

38  a^  x^  +  57  a*  x\ 

19. 

x'-a?. 

20. 

X"  -  a;»  +  2. 

21. 

2  a"  +  4a«+^  —  6a''+\ 

22. 

3x'y-6xUf  +  12x'y'. 

23. 

1  aH^  +  Uan^~28a^b. 

24. 

4a;y-24a;2y2_-36x3y3. 

25. 

5  b^  -  10  ^2  +  15  b. 

26. 

9xy  —  21 X-  y-  +  45  x^  y^. 

80  ALGEBRA. 

Case  II. 

87.    When  one  Term  of  a  Trinomial  is  equal  to  twice  the  Product 
of  the  Square  Roots  of  the  other  two. 

1.  Find  the  factors  of  a'  -\-  2  a  0  -{-  b'. 

a-  +  iah  +  h--  (a  +  b)  (a  +  b) 
We  resolve  this  into  its  factors  at  once  by  the  converse  of  the  prin- 
ciple in  the  Theorem  in  §  74. 

2.  Find  the  factors  of  a'^  —  2ab  +  b'\ 

a-  —  '2ab  +  b'  =  (a  —  b)  (a  —  b) 
We  resolve  this  into  its  factors  at  once  by  the  converse  of  the  prin- 
ciple in  the  Theorem  in  §  75.     Hence  the  following 
Rule. 

Omitting  the  term  that  is  eqiial  to  twice  the  jJroduct  of  the 
square  I'oots  of  the  other  two,  take  for  each  factor  the  square 
root  of  each  of  the  other  two  conneeted  by  the  sign  of  the  term 
omitted. 

Find  the  factors  of 

3.  a-  -h2ab  +  b-.  8.   81  a;2  +  47/2  _  36  x  y 

4.  a-  +  6  a  ^»  +  9  h\  9.    25  a'' ¥  —  \0  ab  c  +  e-. 

5.  a2  _  6  «  i  +  9  b\  10.    X'  —  2abcx-\-  aP'  ¥  c-. 

6.  X-  — lOxy  +  25//-.  11.    ^abxy  +  a-x'^  +  ^h^  >/. 

7.  4  X'  +  12  a;  //  +  9  g'.  12.    1  -  4  x  +  4  a;-. 

Case  III. 

88.    When  a  Binomial  is  the  DifTerence  between  Two  Squares. 

1.    Find  the  factors  of  a-  —  l/-^. 

«2  -b^=(a  +  b)  (a-b) 
We  resolve  this  into  its  factors  at  once  by  the  converse  of  the  prin- 
ciple in  the  Theorem  in  §  76.     Hence  the  following 

Rule. 

Take  for  one  of  the  factors  the  sum,  and  for  the  other  the 
difference,  of  the  square  roots  of  the  terms  of  the  binomial. 


FACTORING. 

Find  tlie  factors 

of 

2.   x^-d'. 

8. 

49  -  c\ 

14. 

x'  -  16  b\ 

3.   x'-l. 

9. 

9-a\ 

15. 

a"  b'  -  9  xK 

4.   4x^-1. 

10. 

x'  -  81. 

16. 

81  a;«  -  25  a^ 

5.   «^-4  6-. 

11. 

9  X-  -  16  y\ 

17. 

121  a^  -  b\ 

6.   9  ic'^  -  /. 

12. 

1  -  a-  b\ 

18. 

ai°  -  a;''. 

7.    1-25  a;-. 

13. 

25  -  64  x^. 

19. 

a«  -  x'. 

81 


89.    When  one  or  both  of  the  squares  is  a  polynomial,  the 
same  method  is  employed. 

1.  Find  the  factors  of  4  x'^  —  (!/  ~  ^)^' 

The  square  root  of  4  x^  =  2  x. 

The  square  root  oi  (y  —  z)^  —  r/  —  z. 

Their  sum  is  2  .r  +  (?/  —  s)  =  2  a;  +  ^  —  s. 

Their  difference  is  2  x  —  (?/  —  s)  =  2  x  -  y  +  z. 

Therefore  4  x^  -  {y  -  z)'^  ^  (2  x  +  y-  z)  (2  x  -  y  +  z). 

Find  the  factors  of  . 

2.  (a  +  by  -  C-.  9.  1  -  (a  -  ^/)^ 

3.  (a  _/,)•■;_  ,.2  10.  (a_2a-)-^-6^ 

4.  .r-  -  (y  +  ;^)-.  11.  (2  .T  -  3  af  -  9  c^. 

5.  x'-(y-zy.  12.  (a  +  ^,)2  _  (,  +  (^)2. 

6.  (x  +  //)■-  -  4  x\  13.  {a  -  by  -  (c-  dy. 

7.  (x  +  2  i/y  -  d\  14.  (4  «  +  .r)2  -  (J  +  yy. 

8.  (a  +  i)'  -  4  c-^.  15.  (5  a;  +  y)^  -  1. 

Resolve  into  factors  and  simplify 

16.  (a  +  3  6)2  -  (a  -  2  by. 

(a  +  3  6)-^  -  (a  -  2  6)2  r=  (a  +  3  6  +  a  -  2  6)  (a  +  3  6  -  a  +  2  6) 
=  (2  a  +  h)  (.5  6) 

17.  (a;  +  yy  -  x\  21.  {a  +  5)^  -  (a  -  6)2. 

18.  x'  -  0/  -  xy.  22.  (2  a  +  1)2  -  (3  a  -  1)^. 

19.  (x  + 3^)2 -4  2/2.  23.  (7x  +  3)2-(5aj-4)2 

20.  (a  -  6)2  -  (a  +  6)2.  24.  («  + 6-3)2- (a-6  +  3)2. 


82  ALGEBKA. 

90.  Polynomials  may  often  be  arranged  in  two  groups  with 
the  minus  sign  between  them,  and  then  factored  as  shown 
above. 

Find  the  factors  of 

1.  X'  —  2x  ij  -\-  if-  —  z^. 

x'^-2xy+  tf  -  ,-.2  =  (x-  ]iy  -z^  =  {x  -ij-{-  :)  (.r  -y-  .:) 

2.  4  ^»2  _  a^  +  2  a  a;  -  x\ 

4  62  -  a2  +  2  a  .r  -  x^  =  4  h'^  -  (cfi  -  2ax  +  x^) 
=  4  b-  -  (a  -  x-y  =  {■2h  +  a~  x)  (-Ih-a  +  x) 

3.  X-  +  2x1/  +  y-  —  dr.  7.  x-  —  or  —  h'-  —  2  a  h. 
Ans.  x  +  II  +  o,  and  x  +  y  —  a.  8.  x-  +  y-  +  2  x  //  —  4  x-  y-. 

4.  «■-  —  2ab+b'^-  X-.  9.  x^  —  x-~2x-^. 

5.  4  a-  +  4  a  ^^  +  ^•-  —  9  C-.  10.  1  -  cr  ~  2  a  b  -  b\ 

6.  2cd  —  l  +  (■-  +  d\  11.  «"  -  1  +  2  X  -  scl 

12.  a^  +  2  a  ^  +  //-  -  <:-  +  2  r  d  -  d'. 

13.  x^+2x  +  l-x--2x-  1. 

14.  a;2  _  4  «  a.  ^  4  «2  _  ^,2  j^^b  y  -  y-. 

15.  1  -  4  a  />  -  a^  +  x--^  _  2  X  -  4  b\ 

91.  Trinomials  of  the  form  x^"  +  x^"  y^"  +  ?/*"  can  be  writ- 
ten as  the  difference  of  two  squares,  and  factored  by  the 
above  method. 

1.  Find  the  factors   a^  +  or  b~  +  b\ 

a*  +  a2  62  +  54  =  a4  +  2  d^  Jfi  +  JA  -  a^  b^ 
=  (a2  +  62)2  ^a^h^=  (a2  +  52  +  a  6)  (a2  +  62  _  a  5) 

Find  the  factors  of 

2.  ««  +  «*  +  1. 

Ans.    a^  +  a  +  1,    a"  —  a  +  1,   and  a*  —  a^  +  1. 

3.  a^  +  rt-  +  1.  6.    a^  -  10  a2  +  9. 

4.  a*-  6  a2  />2  +  //.  7.   a;^  -  3  x^  +  1. 

5.  «»   +    a*  ^-4   ^.   ^,8^  ,S.      X*  -   11  ^2  y2   ^   y4. 

Alls,    x^  —  ^^  +  3  0"  ?/,    x^  —  y^  —  S  X  y. 


FACTORING.  83 

Case  IV. 

92.    When  the  Polynomials  can  be  arranged  in  Groups  of  two  or  more 
Terms,  having  a  Factor  common  to  all  the  Groups. 

1.  Find  the  factors  of  ax  —  a  y  ■\-  h  x  —  h  y. 

ax  —  ay  -\-  bx  -  by  =  (ax  -  ay)  +  (b  x  —  h  y) 
=  a(x-y)  +  b  (x  -  y)  =  (x  -  y)  (a  +  b) 
Or  ax  -  ay  +  bx  -  by  =  (ax  +  bx)  -  (ay  +  b  y) 

=  x(a+b)-y(a  +  b)  =  (a  +  b)  (x  -  y) 

Hence  the  following 

Rule. 

Group  the  terms  of  the  expression  so  that  each  group)  shall 
have  a  monomial  factor,  then  factor  each  group  according  to 
Case  I.,  and  finally  divide  hy  the  factor  common  to  all  the 
groups. 

This  common  factor,  and  the  qiiotient  obtained  hy  the 
division,  will  he  the  factors  required. 

Find  the  factors  of 

2.  ac  +  be  +  ad  +  bd.  7.  x^  —  x'^  y  —  x  y"^  +  y^. 

3.  ar-  be  -ad  +  bd.  8.  x^  +  2  x'  +  4.x  +  8. 

4.  x^  +  a  X-  +  a-  X  +  a^.  9.  x^  —  3  a;-  —  9  x  +  27. 

Ans.     X  +  a,    xr  +  cr.  Ans.   x  —  3,  x  —  3,  a;  +  3. 

5.  x^  —  a  X'  +  ci^  X  —  a^.  1.0.    6x--\-3xy  —  2ax  —  ay. 

6.  x^  +  x^  y  —  X  y'^  —  y^.  11.    a  x  y  +  b  e  x  y  —  az  —  bez. 

Ans.    X  +  //,    x^  —  if.  Ans.     a  -{-  b  c,    xy  —  z. 

12.  X-1  +  x^  -  x\ 

13.  x^  +  a  x^  +  a-  x^  +  a^  x^  +  a*  x  +  a^. 

Ans.     x  +  a,    x~  -\-  a  X  -\-  a^,    x'^  —  ax  -{■  d^. 

14.  x^  -2x*  -4  .r3  +  8  £c2  +  16  a:  -  32. 

Ans.     a;"  -  4  x-  +  16,    x  —  2. 

15.  a""   f  3d^b  +  3  a  //  +  b^. 

16.  «3  —  3dH  +  3ab''  —  b\         Ans.     a  ~  b,    a  —  b,    a  —  b. 


84 


ALGEBRA. 


Case  V. 

93.    A  Trinomial  in  the  form,    .'-'  4-  («  +  ^>)  ■<'  +  «  h,    can  be 
separated  into  two  Binomial  Factors. 

From  the  converse  of  the  Theorem  in  §  78, 

X-  +  {a  +  b)x  +  ab  =  (x  +  a)  {x  -\-  b)  .  .  .  (1) 

X-  —  {a  +  b)  X  +  ab  =  {x  —  a)  {x  —  b)  .  .  .  (2) 

X'  —  (a  —  b)  X  —  ab  =  (x  —  a)  (x  +  b)  .  .  .  (3) 

X-  -{-  (a  —  b)  x  —  ab  —  (x  -{-  a)  (x  —  b)  .  .  .  (4) 

x^  +  13  a;  +  40  =  (x  +  8)  (x  +  5)  .  .  .  (5) 

a;2  -  13  »  +  40  ^  (x  -  S)  (x  -  5)  .  .  .  (6) 

X-—    3  a;  -  40  =  (x  —  8)  (a-  +  5)  .  .  .  (7) 

a;-  +    3  X  —  4.0  =  (x  +  S)  (x  —  5)  .  .  .  (8) 

By  inspecting  the  above  results,  we  find  that,  when  a 
trinomial  is  in  this  form, 

1.  The  first  term  of  both  factors,  in  the  trinomial,  is  the  square  root 
of  the  first  term  of  the  trinomial. 

2.  The  second  terms  of  the  factors  are  such  numbers  that  their 
product  always  equals  the  last  term,  and  their  sum  the  coefficient  of  the 
second  term. 

Hence,  for  factoring  a  trinomial  of  the  form  x^  +  («+&)  x-^ab, 
we  have  the  following 

Rule. 

Find  two  numbers  S2ich  that  their  product  shall  equal  the 
last  term  of  the  trinoinial,  and  their  sum  the  coefficient  of  the 
second  term ;  join  each  number,  respectively,  with  its  proper 
sign,  to  the  square  root  of  the  first  term  for  the  factors 
required. 

1.    Find  the  factors  of  a;"  +  7  x  +  10. 

The  second  terms  of  the  factors  must  be  such  that  their  product  is 
+  10,  and  their  sum  +  7.  The  only  pairs  of  integral  numbers  that 
multiplied  together  make  +  10  are  ±  10  and  db  1,  ±  5  and  db  2.  From 
these  we  are  to  select  that  pair  whose  sum  is  +  7.     These  are  5  and  2. 

The  first  term  of  both  factors  is  the  square  root  of  x^. 

.'.   a;2  +  7  a;  +  10  =  (x  +  5)  (,r  +  2) 


FACTOKING.  85 

2.  Find  the  factors  of  x'-  -\-  x  —  12. 

The  only  pairs  of  integral  numbers  that,  multiplied  together,  make 
—  12  are  ±  12  and  T  1,  ±  6  and  T  2,  i  4  and  ^  3.  The  pair  whose 
sum  is  +  1  is  +  4  and  —  3.     The  square  root  of  x-  is  x. 

.  • .   X-  +  a;  -  12  =  (x  +  4)  (x  -  3). 

Find  the  factors  of 

3.  x'^  -\-  5x  +  6.  17.  9 a;  —  10  +  x'. 

4.  X-  4-  T  a-  +  12.  18.  x-^  —  35  +  2  a;. 

5.  X'  -\-  () X  +  5.  19.  x^  +  7  ax  -\-  10  a". 

6.  a;2  +  10  X'  +  21.  20.  x-  +  12  a  x  +  11  d\ 

7.  x^  J^2x-3.  21.  2  X-  -x-1.     Write  it 
S.  x^-2x-3.  _  (1  +  ^  _  2  x"). 

9.  a;2  _  4  a;  _  J  2.  22.  2  +  x-  x\ 

10.  ic-  —  5  X  +  (i.  23.  110  —  x  —  x". 

11.  a;-^  -  8  a;  +  12.  24.  a;-  +  a  a;  -  42  a-. 

12.  x^  —  7  X  +  6.  25.  a;-^  +  a  a;  -  6  a'. 

13.  a;-^  -  4  a:  -  5.  26.  a'  P  -  3  a  b  c  -  10  c\ 

14.  a;"-  +  4  a:  —  5.  27.  a;-  //-  +  a;  //  ;v  —  12  s;^. 

15.  x'  —  x  —  20.  28.  a  a;2  —  11  a  .r  +  30  a. 

16.  a;2  +  x  -  20.  29.  a:^  +  9  a:^  +  14  x. 

94.  In  the  examples  just  given  the  coefficient  of  the 
highest  power  is  unity ;  but  when  the  coefficient  of  the 
highest  power  is  not  unity,  we  can  often  find  by  inspection 
the  binomial  factors  of  a  trinomial. 

1.    Find  the  factors  of  3  a^^  ^  7  x  +  2. 

It  is  evident  that  the  first  term  of  one  of  the  binomial  factors  is  3  x, 
and  of  the  other  x ;  and  that  the  seccnid  term  of  one  must  be  2,  and  of 
the  other  1,  Since  the  signs  of  7  a;  and  of  2  are  both  +,  the  signs  of 
the  last  terms  of  both  binomial  factors  must  be  4-.  The  factors  must 
therefore  be  either  3x4-2  and  x4-l,  or  3x-fl  and  x  -f  2.  By  trial 
we  find  that  it  is  the  second  set  that  gives  for  the  middle  term  +  7  x. 
Hence  the  factors  are  3  x  4-  1  and  x  +  2. 


86  ALGEBRA. 

2.  Find  the  factors  of  2^^  -  5x  +  3. 

The  first  terms  of  the  binomial  factors  are  2  x  and  x,  and  since  the 
last  term  of  the  trinomial  is  +  and  the  middle  term  is  — ,  the  last  terms 
of  the  binomial  factors  are  both  — .     Hence  the  factors  must  be  either 

2  a;  —  1  and  a;  —  3,  or  2  a;  —  3  and  c  —  1 .  By  trial  we  find  2  a;  —  3 
and  a;  —  1  are  the  factors, 

3.  Find  the  factors  of  ^x-  —  1  x  —  6. 

The  first  terms  of  the  binomial  factors  are  3  x  and  x,  and  the  last 
terms  (disregarding  the  signs)  are  either  6  and  1,  or  3  and  2.  As  the 
sign  of  the  last  term  of  the  trinomial  is  — ,  the  signs  of  the  last  terms 
of  the  binomials  are  different.     Hence  the  factors  of  the  trinomial  are 

3  X  -  6  and  a;  +  1,  or  3  a;  +  1  and  a;  —  6,  or  3  x  —  3  and  a;  +  2,  or 
3  a;  -h  2  and  x  —  3.  It  is  only  the  last  set  that  will  give  —  7  a;  for  the 
middle  term.     Hence  3  a;  +  2  and  a:  —  3  are  the  factors  sought. 

If  the  3's  were  in  the  same  binomial  factor,  the  middle  term  would  be 
divisible  by  3.  Since  then  the  3's  cannot  be  in  the  same  binomial  fac- 
tor, it  can  be  seen  that  neither  3  x  —  6  nor  3  x  —  3  can  be  factors.  This 
at  once  rules  out  two  of  the  four  sets  named  above. 

Find  the  factors  of 

4.  10a;2-7£c-l.  15.   3a;-^- 5.^  +  2. 
Ans.    5a;  -  1  and  2x  -1.       16.   3a;-  -  7a;  +  2. 

5.  3a;2  +  2a;-8.  17.   2a;2  +  a;  -  1. 
Ans.   a;  +  2  and  3a;  -  4.       18.   2a;-  —  3a;  +  1. 

6.  3a:- -fa; -2.  19.   5a;2-3a;-2. 

7.  2a;-  +  3a-  +  1-  20.   hx"  +  9a;  -  2. 
Ans.    2a;  +  1  and  a;  +  1.       21.    Zx"  +  2a;  -  1. 

8.  2a;2+5a;  +  2.  22.   7a;2  +  4a;-3. 

9.  3a;2+  10a;  + 3.  23.    7a:2-20a;-3. 

10.  2a;2  +  3a;-2.  24.  7a;- +  20a; -3. 

11.  3a;- -13a; -10.  25.  7a;2-10a;  +  3. 

12.  5a;2-22a;  + 21.  26.  7 a;^  +  10 a; -f  3. 

13.  5x2-32x-21.  27.  7a;2-9a;-f2. 

14.  5a;2  + .38a;  +  2].  28.  7a;2-5a;-2. 


FACTORING.  87 

Case  VI. 

95.     When  the  Expression  is  a  Binomial  of  the  Form,  «"  ±  &",  n 
being  a  Positive  Integer. 

By  actual  division  : 

a-  +  ah  +  h- 
(3)  '^^^  =  a'  -ah  +  V- 

rt^  -  a3  5  +  fl-2  J2  _  o  ft3  +  J4 

a  +  6 
96.    By  trial  we  shall  find  that 

(1)  a  +  6  is  a  factor  of  a"  +  6"  when  n  is  odd,  but  not  when  n  is  even. 

(2)  a  +  &  is  a  factor  of  a"  —  6"  when  n  is  even,  but  not  when  n  is  odd. 

(3)  a  —  ft  is  a  factor  of  a'*  —  ft"  always. 

(4)  rt  —  ft  is  a  factor  of  a"  +  6"  never. 

Note.     One  can  always  answer  any  one  of  these  four  points  by  consider- 
ing the  matter  thus  : 

(1)  Willa  +  6  divide  n -I- 6?     Yes.     Will  a  +  6  divide  a2  +  62  ?     Nq. 

(2)  Will  a  +  h  divide  n  -  6  ?     No.      Will  a  +  b  divide  cfi  -  Ifl  ?     Yes. 

(3)  Will  a  -  6  divide  a  -bl     Yes.     Will  a  -  h  divide  (/-  -  6^  'i     Yes. 

(4)  Will  a-b  divide  «  +  6  ?     No.      Will  a  -  b  divide  cfi  +  b'^  1     No. 

Attention  to  the  following  laws  will  enable  one  readily  to 
write  out  the  factors  of  such  expressions. 

1.  The  terms  of  tlie  quotient  are  all  jwsitive  when  the  divisor  is 
a  —  ft,  and  alternately  positive  and  negative  when  the  divisor  is  a  +  b. 

2.  The  number  of  terms  corresponds  to  the  degree  of  the  binomial. 

3.  a  appears  in  the  first  term,  6  in  the  last  term,  and  a  and  ft  in  all 
the  intermediate  terms. 


a 

-  ft 

n^ 

-ft3 

a 

-  ft 

a^ 

-ft* 

a 

-ft 

«2 

-ft2 

a 

+  ft 

a* 

-ft* 

a 

+  ft 

rt3 

+  ft3 

a 

+  ft 

a- 

+  ft5 

88  ALGEBRA. 

4.  The  exponent  of  a  in  the  first  term  is  one  less  than  the  degree  of 
the  binomial,  and  decreases  regularly  by  unity  in  each  successive  term  ; 
the  exponent  of  b  in  the  second  term  is  1,  and  increases  regularly  by  one 
in  each  successive  term,  till  in  the  last  term  it  becomes  the  same  as  the 
exponent  of  a  in  the  first  term. 

5.  The  sum  of  the  exponents  of  a  and  b  in  any  term  is  alwaj's  the 
same,  and  is  equal  to  the  exponent  of  a  in  the  first  term,  a  and  b 
stand  for  any  letters  or  expressions. 

Find  the  factors  of 


1. 

x'  +  f. 

x^  +  y^ 

=  {x  +  y)  (x^  ■ 

-xy  +  y^) 

2. 

x'  -  y\ 

x^-f 

=  (x-y)  (x^ 

+  xy  +  y^) 

3. 

x>  +  yK 

x^  +  r/ 

=  (x  +  y)(x* 

-.vSy  +  x^y^-xy^  +  7/) 

4. 

c'  -  8. 

c3-8 

=  C3  -  23  =   (C 

-  2)  (c2  +  2  c  +  22) 

5. 

f/  -  IjK 

14. 

x"  -  32. 

23.    125a3-l. 

G. 

x'  +  y\ 

]5. 

x'"  +  64. 

24.    Sa^-r^  d\ 

7. 

c^-1. 

16. 

x^  -  64. 

25.    x^-y\ 

8. 

c^  +  1. 

17. 

6x'-  48. 

26.   27y/-a;«. 

9. 

x«  +  8. 

18. 

27  a'  -  r\ 

27.    an^c^-iP. 

10. 

\-x\ 

19. 

«3  _  343. 

28.    a^l>\-'-l. 

11. 

x""  +  27. 

20. 

a'^  -  243. 

29.    54a«-2. 

12. 

x^  -  27. 

21. 

125  -  a\ 

30.   24:a^-Slb\ 

VI 

x^  -  a\ 

22. 

a^  -  125. 

31.    32a;^-108rc?> 

97.  When  in  a"  —  h"  n  is  even  and  greater  than  2,  there 
will  be  three  or  more  factors  in  each  case,  and  they  can  be 
more  expeditiously  determined  by  Case  III.,  with  other  prin- 
ciples already  explained. 

Find  the  factors  of 


b\ 

-Mr: 
-6« 

:  (a^ 

f  h-^)  (a^  - 

-b'^)  = 

(a^  + 

V')  («  + 

b)(a- 

-b) 

=  ((r 

5  -t-  h^)  (a^ 

-/.3) 

=  (« 

+  h)  («2 

-  a  h  + 

b-')  (a 

-  b)  (n^ 

+  ab  +  h^) 

3. 

l-a\ 

FACTORING. 

l-a8  = 

:(1   + 

o4)(l  -a*)  =  {l  +a* 

)(!  +  ' 

r^)  (1  -  a^) 

:(1    + 

a*)(l  +  a')  (1  +  a)  (] 

L-«) 

4. 

««  -  b\ 

11.    a'-l. 

18. 

a}^  -  a\ 

5. 

x'  -  if. 

12.    aio  -  ^1". 

19. 

o"  -  a\ 

6. 

a'-l. 

13.    a'-W. 

20. 

a«  -  64. 

7. 

l-a\ 

14.   «*-81. 

21. 

16  x^  -  X. 

8. 

fl«-l. 

15.    81 -a^ 

22. 

a}'  -  a\ 

9. 

l-««. 

16.   a.Uj*-c\ 

23. 

a}^  />i^  -  a' 

:o. 

1  -a\ 

17.  x^-^-y^. 

24. 

^10  ^10  _  ^i 

Fi 

nd  the  factors  of 

1. 

x'  +  2/«. 

a.6  +  y6  =   (3.2)3  +    (^yiy  =   (.J.2 

-(x-^ 

2. 

a;^2  +  y'\ 

3. 

xi°  +  ?/^°. 

4. 

l  +  x«. 

98.  Though  «"  +  h"  is  not  divisible  by  «  +  6  when  n  is 
CVC71,  it  is  possible  to  find  a  binomial  factor  in  every  case 
except  when  n  is  a  power  of  2,  as  2,  4,  8,  16,  etc. 


^2)  1^(^2)2  _  ^2^2  +(^/-2)2] 
if)    (X"  -  X2  //2  +  2/4) 

5.  a^"  +  1. 

6.  1  +  x^\ 

7.  a;«  if  +  ;5«. 

99.    To  be  expert  in  factoring,  it  is  necessary  to  become 
familiar  with  the  following  algebraic  expressions : 

a^  +  l>^       is  prime. 

a2  _  Ifi  ^  (a  +  6)  (a  _  h). 

«3  _^  6^  =  (a  +  b)  (a'  -ah  -\-  b^). 

a^  -b^^ia-  b)  (a-  +  ab  ^  b""). 

a'*  +  b'^       is  prime. 

a^  _  />4  =  («-^  +  /.^)  (.y  +  b)  {a  -  b). 

o!"  +  b^=  (a  +  b)  (a*  -aH  +  aH''-  a  b^  +  b*). 

a'-b'  =  (a-  b)  (a"  +  a^  h  +  a-  b'  +  «/>'  +  ^''). 


90  ALGEBRA. 

a^  +  b^=  {a"  +  b^)  (a*  -  a"  b^  +  b"). 

a^  —  W={a^  b)  {d^  -ab  +  b-)  (a  -  b)  («-  +  ab -\-  b^). 

a'  +  ^7  ^  (a  +  b)  (a«  —  aH  +  a*  b-  ~  a" //  +  a- b^  —  aP  +  b^). 

a'  -  b''  =(a-  b)  (a"  +  ct'  b  +  a"  h'  +  a^  /y=^  +  n^  b^  +  ah"^  +  b^), 

a^  +  b^       is  prime. 

a«  -  b^  =  (a^  +  /.4)  {d'  +  ^>-^)  (fl  +  b)  {a  -  b). 

a^  J^2ab  +  ¥    =  {a  +  bf. 

a2  _  2  «  6  +  ^,2     ^  (^  _  ^y2_ 

«*  +  2a2^<-^  +  ^,4^  (,r^  +  /,2|2_ 

a4  _  2  rr^A-  +  b^  =  (a'  -  b-f  =  (a  +  by  (a  -  by. 

n^  +  2aH^  +  b''  =  (a^  -\-  b'y  z=  (a  +  by  (a'  -  a  b  +  U'y. 

a^-2aH^  -^  U'  =  (r/^  _  Jfiy  =  («  _  by  (a}  +  a  b  -^  b'y. 

100.    Division  by  Factors, 

Divide 

1.  a2  _  ^2  l^y    ^_Jj  15^      ,,3  _   ^3   |,_^.   ^,2   ^_    ,,  ft   +    ft2_ 

2.  a3  _  ft3  Ijy    ^j  _  ft_  16       ,,3   _|_    ft3   b^.    ^^2  _    o  ft   +   ft2. 

3.  ,,3  _^  ft3  |5y  „  ^  ft^  17_  9  -  a;2  by  3  -  a-. 

4.  r/S  +  fts  by  a  +  b.  18.  .T^  —  ?/^  by  x^  +  x"^/  +  2/*- 

5.  ftS  _  ft5  ]3y  ^,  _  ft,  19,  27  +  x^  by  3  +  X. 

6.  «  _  b  by  /y  -  r/.  20.  ;/■«  -  if  by  ^r^  +  y\ 

7.  ^2  _  1  by  (I  +  1.  21.  X-  +  7  .)■  +  12  by  x  +  4. 

8.  r/8  -  1  by  f/  -  1.  22.  x^  -lx+  12  by  x  —  A. 

9.  a^  +  1  by  a  +  1.  23.  a"  +  2  a  b  +  V  by  (/  +  b. 

10.  1  -  a3  i^y  1  _  ^,,  24.  a^  +  2  a  +  1  by  «  +  1. 

11.  4  ar2  _  9  y2  by  2  ar  -  3  ?/.  25.  a^  _  2  r:  /;  +  W  by  «  -  b. 

12.  1  -  9  a;2  by  1  -  3  X.  26.  a^  -  2  r.  +  1  by  a  -  1. 

13.  IG  .x"  _  y^  by  4  x2  -  7/2.  27.  x'^  -  G  x  +  5  by  £k  -  5. 

14.  125  +  <i^  by  5  +  </.  28.  a;'^  +  9  x  —  10  by  x  —  \. 

29.  x2  -  G  a  a;  +  9  a2  by  a:  -  3  r?. 

30.  x^  +  5  a;  ?/  -  3G  ?/2  by  x  -  4  y. 


FACTORING. 


91 


31.  (x  +  ly  (a;  +  2)  by  a:  +  1. 

32.  (x  -  1)  (a:^  -  9)  by  ic  -  3. 

33.  (,,2  _  0-^)  (a  -  b)  by  r?.-^  -  2  a  b  +  ^-^ 

34.  {a  +  ^')'  -  ^-  by   a  +  h  +  c. 

35.  «3  _  3  rt'^  ^^  +  3  a  b-  -  b^  by  a''  -  2  a  b -\-  1?. 

36.  a"  +  r/2  />^  +  lA  by  <r^  +  ^'  />  +  /A 

37.  ci"  +  //-  +  <■-  +  2  r/  />  +  2  r,  r  +  2  i  c  by  a  +  ^'  +  c. 

38.  «"  +  />i-  1\\-  rt^  +  b\ 

39.  a-  +  r.  ^^  +  ^'  ^  +  />  c  by  a  +  ^>. 

40.  o?  —  ac  -\-  a  b  —  b  c  by  «  —  c. 


101.     MISCELLANEOUS    EXAMPLES. 

Factor  the  following : 


1.  a:*_a---42. 

2.  a;-  -  9  a;  y  -  10  y". 

3.  x"  —  xK 

4.  6^  +  1  +  9x1 

5.  a-^  —  8  X  ?/  +  7  y^. 
0.  a-13  —  x^. 

7.  a-  a;  4-  ^/  a-  —  </  —  r^^. 

8.  a-*^  +  2  a;-  +  4  a;  +  8. 

9.  a'-  —  '/'^  +  }f  —  2xy. 


10.  rr^  -l  +  b-'-2  a  b. 

11.  16a;^-l. 

12.  2  a"  -  2  a  b\ 

13.  7  a;- -50  a;  +  7. 

14.  2  r^  a;^  -  22  a  x'  +  60  a  x. 

15.  2r/2_a_15. 
ir>.  ,r^  +  5r/a-  +  6x'-^. 

17.  a:*  //  —  X-  y^  —  x^  y-  -\-  xy'^. 

18.  x'^  —  if  +  X  —  y. 


19.  2bc  +  a'  +  2ad-  c'  +  cP  -  Z*'-', 

20.  a^  +  b-'  +  r--^  +  2  ^?  Z- 

21.  (a  +  by  +  a  +  Z*. 

22.  a:8  +  x*y*  +  7f. 

23.  x^-4: 7f  +  .X  +  2  2/. 


24.   2xy  +  4 


25.    a--^ 


-2a-  +  1. 


2ac  —  2bc. 

26.  2a;2  +  a^-l. 
r/^  +  a^  +  a. 
a^  +  3a^  +  3a  +  1. 

/3  rt»2  ___    ,,3  ™*2 ,,3  ^,2 


^r  a;' 


"  r +  ^  r 


92  ALGEBRA. 


CHAPTER    XII. 
GREATEST    COMMON    DIVISOR. 

102.  A  Common  Divisor  of  two  or  more  algebraic  expres- 
sions is  an  expression  which  will  divide  each  of  them  without 
remainder. 

103.  The  Greatest  Common  Divisor  of  two  or  more  algebraic 
expressions  is  the  expression  of  the  highest  degree  which  will 
divide  each  of  them  without  remainder. 

104.  From  this  definition  it  is  evident  that  the  greatest 
common  divisor  of  two  or  more  algebraic  expressions  must 
contain  all  the  factors  common  to  the  expressions,  and  no 
others ;  and  a  factor  may  be  introduced  into,  or  rejected  from, 
one  of  two  algebraic  expressions,  if  it  eontains  no  factor  of  the 
other,  without  affecting  the  greatest  common  divisor. 

Note.  The  names  (jreatest  common  measure,  highest  common  measure,  high- 
est common  divisor,  highest  common  factor,  ase  used  by  different  autliors  to 
mean  the  same  thing  as  greatest  common  divisor. 

Case  I. 

105.  To  find  the  Greatest  Common  Divisor  of  Monomials  and 
Polynomials  which  can  be  resolved  into  Factors  by  Inspection. 

1,  Find  the  greatest  common  divisor  of  20  x'- y'^  z'^,  2ix'^i/, 
and  2Sx'fz\ 

20a^-=/.t;''  =  22.  5x^fz* 
2Ax^y       =2^-Sx^'y 

.-.   G.  C.  D.    =2^  -x^y 

It  is  evident  that  the  highest  power  of  2  which  will  divide  all  three 
expressions  is  2^  ;  of  x,  x^ ;  of  y,  y  ;  and  that  ::  will  not  divide  them  all  ; 
tlierefore  the  greatest  common  divisor  is  2'^x'^y. 


GREATEST   COMMON   DIVISOR.  93 

2.  Find  the  greatest  common  divisor  of  a^  +  ^'^    f^'?'  —  6^  and 

a^  +  a  b. 

a^  +  />3  =  (,,  ^  ^^)  (^-2  _  ^,  ^^  _,_  ^o-) 

a2  _  ^-2  ^  (,,  +  5)  (a  _  ^) 

a^  +  <*  ^  —  *  ("  +  ^) 
.-.   G.C.B.^a  +  h 

From  these  examples  we  derive  the  following 

Rule. 

Separate  each  expression  into  its  prime  factors ;  then  take 
every  factor  common  to  the  given  expressions  the  least  number 
of  times  it  occurs  in  any  one  of  them  for  the  greatest  common 
divisor  required. 

Find  the  greatest  common  divisor  of 

3.  a^b^  <iH.  12.  3«V«,  21  a^  r\  ISa^i-r". 

4.  abc^  bed.  13.  a- -\- ax,  a'-^  —  ax. 

5.  2  ax,   3  x\  14.  d-  +  2ab,   ab  +  2  b-. 

6.  6  a  be,   lb  bed.  15.  (x  +  a)-,   {x  +  af. 

7.  aHr,   ab'r,   abc^  16.  {^  {a  +  b)\   15  {a  +  b)\ 

8.  4  a  c^  8  a2  e"^,  12  «  b  r.  17.  2  («  -  bf,  6  (a'^  -  i^). 

9.  12  (a;-^  -  9),  S{x-  3)^  18.  a;'^  -  16,  a;"^  +  4  a-. 

10.  cc-  +  4  a;  +  3,  a;-'  —  1.  19.    a  ,x  +  7  o,  a;"''  +  a;  —  42. 

11.  2  a;2  +  o  .c  —  3,   a;2  -  9.        20.    x*  —  y\  {x  —  yf  {x  +  yf. 

21.  a^  +  a;^   a'^  —  a;■^  a'^  +  2  r<  a-  +  a;^ 

22.  a;'^  —  y^,  x^  ■\-  xy,  x^  y  -\-  xy^. 

23.  x}  —  x  —  6,   a;-  +  3  a;  —  18. 

24.  a;«  -  a*',   (a;'^  +  «  a;  +  «')  (a-'-^  +  a'). 

25.  a;«  +  a«,    (a;"  —  a'^T^  +  «*)  {x''  —  a-). 

26.  (a;-y)^    {x''-y^){x-y)\ 

27.  2a;2  -  5a;  -  3,  x"  -  8a-  +  15. 

28.  3a;-'^  -  5a:  -  2,    6a-2  +  5  a-  +  1. 


94  ALGEBRA. 


Case  II. 


106.    To  find  the  Greatest  Common  Divisor  of  Polynomials  which 
cannot  be  factored  by  Inspection. 


After  removing  every  monomial  factor  j^ossible  from  each 
expression,  arrange  the  resulting  erjjressions,  according  to  the 
descending  powers  of  some  common  letter,  and  divide  the  ex- 
pression which  is  of  the  higher  degree  by  the  other.  Continue 
the  division  until  the  remainder  is  of  a  loivcr  degree  than  the 
divisor.  Then  make  the  remainder  a  nev^  divisor  and  the 
divisor  a  new  dividend ;  and  continue  the  process  until  there 
is  no  remainder.  The  last  divisor,  together  with  the  common 
monomial  factors,  removed  at  the  beginning  of  the  o2)eration, 
will  be  the  greatest  common  divisor. 

Note.  Monomial  factors  should  be  rejected  when  possible,  and  introduced 
only  to  avoid  fractious  (§  104).  If,  after  the  removal  of  such  factors  at  any 
point  of  the  process,  polynomials  of  tlie  same  degree  appear,  either  may  be 
used  as  the  divisor,  though  it  is  better  to  take  as  the  divisor  tlie  one  whose 
first  term  has  the  smaller  coefficient. 

1.  Find  the  greatest  common  divisor  (G.  C.  D. )  of  x^  +  x'^  —  2, 
and  a;3  +  2x'-rK 

3,3  _^  ^.2  _  2  )  ,,.3  +  2  ^2  _  3  (  1 
x^  +     x^~2 

x"^  -  \  )  x^  +  x^  —  2  {x  +  1 

X^  —  X 


x^-^x  -  2 
a;2  -1 


G.  C.  D.  rr  X  _  1  x-l)x'-\{x-\-l 

^•2  _  -. 


GREATEST   COMMON   DIVISOR.  95 

2.    Fiiid   the   G.  C.  D.   of   24  x*  -  2  x^  -  60  x-  -  32  x,   and 

The  following  arrangement  saves  rewriting  the  divisor  when  it  becomes 
the  dividend  : 


x)24a:*- 

2x3 

-  60  X  2  - 

-32x 

12x3- 

x-^ 

-30x- 

16 

12x3- 

4x2 

-26x- 

12 

3x2 

-    4x- 

4 

3x2 

+    2x 

-6x- 

4 

-6x  - 

4 

3x)  18. 


6  x3  -  39  x2 


6  x3  -  2  x2 
6  x3  -  8  x2 


13X-6 

8x 


6  x2 
6x2 


5x  —  6 

8x-8 


3x  +  2 
G.  C.  D.  =  X  (3  X  +  2) 


3.    Find    the    G.  C.  D. 
6x3  +  ^2_44_^^  21. 


of 


x'- 13x- +  23a;-21,      and 


X 

3x3 
3x3 

-  13x2  +  23  X-  21 

-  10x2+     -J. 

6x3+       x2-44x  +  21 
6x3-  26x2+  46X-42 

1 

-3x2+  16X-21 
-3x2+  10.^-    7 

9)27x2-90x  +  63 

3x2-10x+    7 
3x2-    7^. 

2)6x-  14 

3x-    7 
=  3  .c  -  7 

G.  C.  D. 

-3x+    7 
-3x+    7 

4.    Find  the  G.  CD.  of  2x2-5.-^  +  2,  and  .T3  +  4a:2-4x-16. 


2  x2  -  5  x  +  2 
2  x2  -  4  X 

x3  +  4x2-4x-  16 
2 

2x 

2  x3  +  8  x2  -  8  X  -  32 
2  x3  -  5  x2  +  2  X 

-' 

-x  +  2 
-x  +  2 

13x2-  lOx-32 
2 

=  x-2 

26  x2  _  20  X  -  64 
26  x2  -  65  X  +  26 

45  )  45  X  -  90 

G.  C.  D. 

X-    2 

13 


96  ALGEBRA. 

Find  the  G.  C.  D.  of 

5.  4.x^-\-'Sx-10,  4:x^  +  7x^-3x-l5.         Ans.  4a;-5. 

6.  Sx^+Ux  —  15,   Sx^  +  30x"  -\-l'3x  —  30. 

7.  a:2  -  7ic  +  10,  4.x^-25x^  +  20a;  +  25. 

8.  x^  +  x^  -\-  X  —  3,  x^  +  3x'  +  5x  +  3. 

Ans.    a;2  ^  2  a;  +  3. 

0.    a;*  —  2  a;-  +  1,   x^  —  4  a;='  +  ti  a;^  —  4  a;  +  1. 

Aus.    a;''^  —  2  a;  +  1. 

10.  a;"^  —  5  a;  ^  +  4  y%   x^  —  5  x^  //  +  4  xy^. 

11.  2x2  —  ox  +  2,   4x^  +  12x-^  —  a;  —  3. 

12.  2x2  -  5xy  +  2/,  4x=^  +  Vlx^y  -  xf  -  3y\ 

Ans.    2x  —  y. 

13.  x=^  +  3  X-  +  4  X  +  12,  x^  +  4  x'-  +  4  X  +  3. 

14.  3x''  -  13x-  +  23x  -  21,  Qx^  +  x^  -  44x  +  21. 

Ans.    3x  —  7. 

15.  4x«  +  12x*y  -  40x2^/2^  2x*  -  6x2;y  +  4/. 

16.  2  a^  -  5  a2  +  2  a,   2  a^  -  3  a'^  -  8  a"  +  12  a. 

Ans.    a  {a  —  2). 

17.  12  b'  -  30  6  +  12,  36  &3  _  24  ^2  _  9  J  +  6. 

Ans.    3(2ft-l). 

18.  X*  -  3x3  ^  2x2,  x"  -  3x2  _,_  2x. 

19.  2  x^  —  X  +  x"  -  2,  2  x2  +  X  +  3  —  x^  +  x*. 

Ans.    x2  +  X  4-  1. 

20.  2x/  -  2rf  +  6x3  —  6x2^,  8x2y  +  2/  —  lOx/. 

Ans.    2  (x  —  y). 

21.  x''  +  4x2  +  16,  2x''  -  x^  +  16x  -  8. 

22.  3  x^  +  5  x2  —  X  +  2,  4  x^  +  9  x^  +  2  x2  —  2  X  —  4. 

Ans.   X  +  2. 

23.  2  a*  +  aH  —  4  a-  b'  —  3  a  b%  Aa*  +  aH  —  2  a^b^  -^  a  P. 

24.  6  X*  +  x3  —  a',  4  x''  -  6  x^  -  4  x2  +  3  x. 

25.  4  .x^  -  18  x'  +  19  X  -  3,   2  x"  -  12  x^  +  19  x2  -  6  x  +  9. 


LEAST   COMMON   MULTIPLE.  97 


CHAPTER    XIII. 
LEAST    COMMON   MULTIPLE. 

107.  A  Common  Multiple  of  two  or  more  algebraic  ex- 
pressions is  an  expression  that  can  be  divided  by  each  of 
them  without  remainder. 

108.  The  Least  Common  Multiple  of  two  or  more  algebraic 
expressions  is  the  expression  of  the  lowest  degree  that  can  be 
be  divided  by  each  of  them  without  remainder. 

109.  It  is  evident  from  the  above  definitions  that  a  com- 
mon multiple  of  two  or  more  expressions  must  contain  all 
the  different  factors  of  these  expressions ;  and  the  least  common 
multiple  of  two  or  more  expressions  must  contain  only  the 
factors  of  these  expressions. 

Case  I. 

110.  To  find  the  Least  Common  Multiple  of  Monomials,  and  Poly- 
nomials which  can  be  resolved  into  Factors  by  Inspection. 

1.  Find  the  least  common  multiple  of  lQa?b^,  12  a^b^c^, 
and  20  aH^c. 

16  a^  b^      =2*  ■  a^  b^ 
12  an'c^  =  2^-  3aH^c^ 
20aH^c   =2^-  ba^b^c 


L.  C.  M.  =  2*  -S-San^c^ 


It  is  evident  that  no  number  which  contains  a  power  of  2  less  than  2*, 
of  a  less  than  a^,  of  b  less  than  6^^  of  c  less  than  c^,  and  which  does  not 
contain  3  and  5,  can  be  divided  by  each  of  these  numbers  ;  therefore  the 
least  common  multiple  is  2*  .  3  ■  5  a^  js  ^.s 

7 


98  ALGEBRA. 

2.  Find  the  least  common  multiple  (L.  C.  M.)  of  4  (a;  —  yY, 
'6{x'-i/),    Q{x'  +  2xy+  i/). 

4.{x-yy  =  2-'ix-yy' 
^{x''-y')=^{x  +  y){x-y) 
Q{x^  -\-  2xy  +  y-)  =2  .3{x  +  y)- 

.'.   L.  C.  M.  =  22  .  3  (x  +  yy  (x  -  yf 
From  these  examples  we  derive  the  following 

Kule. 

Separate  each  expression  i7ito  its  75?'me  factors,  and  then 
take  every  factor  the  greatest  numler  of  times  it  occurs  in 
any  one  of  the  expressions  for  the  least  common  multiple 
required. 

Find  the  least  common  multiple  of 

3.  4  a^,   6  a^.  15.  a(x  +  a),  b(.r  +  a),  c(x  +  a). 

4.  Qx^y,   15xy^.  16.  x^  —  y"-,   (x — y)^. 

5.  ah,   ac,   b  c.  17.  4:  (x  —  y),   x^  —  y^. 

6.  10a;^   Qx'y^   V2x^y.  18.  x"^  -  x,  x'^  —  1. 

7.  ccS  4  x3 ?/,  6  x^ /,  4  X  f,  y\  19.  x-  —  d\  (x  +  a)-,  (x  —  af. 

8.  ay,   az\   a^z,  az.  20.  {x  +  If,  4.(x^-\),   6. 

9.  dx,  3(a-a;).  21.  x(x-y),  x(x^-y^),  x. 

10.  abc,  ab(a-  c).  22.  x  +  y,  x^  -  y%  x^  +  /. 

11.  2a^(a-{-x),  4.ax.  23.  x  -  3,   x -\- 3,  x'^  —  9. 

12.  3  (a  +  b),  7  (a+  b).  24.  ax  +  by,  ax -by,  a^x^-bh/. 

13.  a^b{x-y\   ab^{x-y).      25.  {x  —  a)\   (x-a)(x-b). 

14.  a^  (x  +  a),   <r  h\  h"-  (x  -  a).  26.  a;^  -  8,  a;^  -  4x  +  4. 

27.  (x  —  yY,  x/-y\ 

28.  x^  —  /,  a^-  —  a;,y  +  ;/-,  x'^  +  xy  +  y^. 
20.    (x  +  3)3.  a;3  +  27. 

30.  x^  —  y"^,  x^  +  y^  x^  —  a;y  4-  .y^- 

31.  a:2 


LEAST   COMMON   MULTIPLE. 


99 


Case  II. 

111.    To  find  the  Least  Common  Multiple  of  Polynomials  which 
cannot  be  readily  factored  by  Inspection. 


Divide  one  of  the  expressions  hy  their  greatest  common 
divisor,  and  multipli/  this  quotient  hy  the  other  expression  for 
the  least  common  midtiplc  required. 

1.  Find  the  least  cominon  multiple  (L.  C.  M.)  of  o;^  —  4a;  +  15, 
x"  +  X'  +  -o. 

4  X  +  15  -r^  +     x^  +  25 

x^  -•6x^  +  bx  x*  -  4  x«  +  15  X 


3  x2  -  9  X  +  15 
3x2-9x+  15 


5  )  5  x^  -  15  X  +  25 

x2-    3x+    5  =  G.  C  D. 


L.  C.  M. 


_  (x3  -  4  X  +  15)  (x*  +  x2  +  25) 

~  x2  -  3  X  +  5 

=  (x  +  3)  (x"  +  x2  +  25) 

or  (x2  +  3  X  +  5)  (.k3  -  4  X  +  15) 

or  (x  +  3)  (x2  +  3  X  +  5)  (x-^  -  3  X  +  5) 


Find  the  L.  C.  M.  of 

2.  x^  4-  6a;  +  8,  x^  +  ^x^  +  1  x  +  2. 

Ans.     {x  -^2){x  -{-  4)  (a--  +  oa-  +  1). 

3.  4  a^  —  5  a  6  +  ^"j   3  a^  —  3  a'-  6  +  ab^  —  ¥. 

Ans.     (4  a  —  b)  {a  —  b)  (3  a-  +  b^. 

4.  x'  -  7a;  +  10,   ^x^  -  25a;2  +  20  a-  +  25. 

Ans.     {x  -b){x-  2)  (4 a;-^  -  5 a;  -  5). 

5.  4a;2  +  3a;  -  10,  4,r3  +  1  x"  -Sx-  15. 

Ans.     (4 a;  -  5)  (a;  +  2)  (a;'^  +  3 a;  +  3). 

6.  x^  -  8a;  +  3,  a;«  +  3a;5  +  a;  +  3. 

Ans.     (cc  +  3)  (a;2  -  3  a;  +  1)  (^'  +  1). 

7.  a;3  -  6a;2  +  11 X  -  6,  x^  -  9a;2  +  26 a;  -  24. 

Ans.     (.X-  -  1)  (a;  -  2)  (a;  -  3)  (a-  -  4). 


100  ALGEBRA, 


CHAPTER    XIV. 
FRACTIONS. 

112.  When  division  is  expressed  by  writing  the  dividend 
over  the  divisor  with  a  line  between,  the  expression  is  called 
a  Fraction.  As  a  fraction,  the  dividend  is  called  the  numera- 
tor, and  the  divisor  the  denominator. 

Thus,  -r-  is  a  fraction  whose  numerator  h  ab  and  denominator  b,  and 
whose  value  is  a. 

113.  From  the  principles  of  division,  it  follows  that 

-{-  ab  _       —  a  b  ^  —  ab  ^  -\-  ah 
-\-  a               —  a              -\-  a  —  a 

,  —  ab  _  +  ab  _  -\-  ab 

+  a  —  a  -\-  a 

That  is,  {a)  The  value  of  a  fraction  is  not  changed, 

(1)  If  the  sign  of  every  term  of  the  numerator  and  denom- 
inator is  changed. 

(2)  If  the  sign  of  every  term  of  the  numerator  and  the 
sign  hefore  the  fraction  are  changed. 

(3)  If  the  sign  of  every  term  of  the  denominator  and  the 
sign  hefore  the  fraction  are  changed. 

(b)  But  the  value  of  a  fraction  is  changed, 

(4)  If  every  sign  of  the  numerator,  or  denominator,  or  the 
sign  hefore  the  fraction,  is  changed. 


FRACTIONS.  101 


114.      EXERCISES. 


Express  the  following  fractions  in  fonr  different  ways  : 
a  —  b       b  —  a  b  —  a  a  —  b 


"•    c-d-d- 

-  c 

-      c-d~      d- 

c 

o       3a 

5x 

6.    '-\ 

IJ-  X 

x-y 

^-    d^-b-^- 

3.    ^-^ 

z 

3x-  1 

\  —  x^' 

a-c 

c  —  a 

^ 

c  —  a 

""■    ia-b){x- 

■a) 

-  (b-a){x-  a) 

{a 

-b){a-  X) 

a  —  c 

~  ip-  «)  («  -  ^) 

Changing  the  signs  of  an  odd  number  of  factors  of  a  prodact 
changes  the  sign  of  the  product ;  but  changing  the  signs  of  an 
even  number  of  factors  of  a  product  does  not  change  the  sign  of 
the  product. 

Write  the  following  fractions  so  that  x  and  y  shall  become 
positive  : 


9. 

—  x            x 

^-y    y-^ 

12. 

a  —  X 
b-y- 

10. 

Z  —  X 

-y 

13. 

a  —  x  —  y 
2^-9     • 

11. 

a  —  X  —  y 
(«  -b){c-d)' 

14. 

z-y 
-x{x''-l) 

REDUCTION    OF    FRACTIONS. 

115.   Reduction  of  Fractions  is  changing  their  form  without 
changing  their  value. 


102  ALGEBRA. 

Case  I. 
116.     To  reduce  a  Fraction  to  its  Lowest  Terms. 

A  fraction  is  in  its  loioest  terms  when  its  numerator  and 
denominator  have  no  common  factor,  that  is,  are  mutually 
prime. 

a  __a  III  a  in       a 

b       h  lit  b  m       b 

That  is,  multiphjiiuj  or  dioidiiuj  both  numerator  and  denom- 
inator hij  the  same  number  does  not  change  the  value  of  the 
fraction. 

Hence,  to  reduce  a  fraction  to  its  lowest  terms,  we  have 
the  following 

Rule. 

Divide  both  terms  of  the  fraction  hy  any  factor  common  to 
them ;  then  divide  these  quotients  by  any  factor  eominon  to  them  ; 
and  so  proceed  till  the  terms  are  mutually  2irime.     Or, 

Divide  both  terms  by  their  greatest  common  divisor. 

Reduce  the  following  fractions  to  their  lowest  terms : 
12  ar'  h'  X 


l^a*b^'< 


12  a'^b^x         6b'-x  2x 


xy 


ISaH-y       9a^b-y       3a^y 
1        „     ax  —  bx 


Ans. 


xy^  y 


abx 


abf  ^             loC'h 

b-c    '  14  a'' +  21  a"  c 

5b*x  f,  a-  ■{■  ab 

20b^x* '  "  ab  +  b"^' 

ab'x  ^^  («-^)' 


2,ah^x}'  a^-b^ 

ax'  ..      a''-b^ 

x^y-x"'  '    a^-Yb^' 


12. 
13. 
14. 
15. 
16. 
17. 
18. 
19. 
20. 
30. 


a^ 

-h^ 

a' 

-W 

X 

—  il 

f 

-x'' 

h^ 

-a' 

(a 

+  bf  • 

X 

9 

4- 

-X^' 

x' 

+  4.xy 

x' 

-  l(^r 

x' 

-1 

FRACTIONS.                                            ] 

21. 

a^  +  2a'x-h  ax" 

22. 

5ax  —  15  a^ 
x^-9  a^     ' 

23. 

x^-5x  +  6 

x^-7x  +  12' 

24. 

'lo. 

cc^  +  4  cc  +  4 

26. 

a-  +  oa  +  Q 

3  n-^  +  2  «  -  8  ■ 

OJ 

a^-b^ 

{a^  -  b^)  {a  +  b)  ' 

28. 

(a  -  by  -  c" 
{a-b  ■{-  cf  ' 

29. 

{c  -  a)  (c  -  b) 

103 


x^-1' 

a^  —  2ax  +  x' 

d^  —  x^ 
2x^-4.x'' 
x^--lx*   ' 
3x^-12  ax 

48  d'  -  3  x'  '  '"'    {a  -b){a-  c)  (b  -  c) 

x^  —  xy  -{-  X  — 
~~x^  +  2  .r  +  1 


(g  +  by  -  (c  +  dy  aJ^h  +  c-\.d 

'    •    (a-cY-id-by  '^"'-     a~b-c  +  d' 

d^  -\-  ab  +  ac  -{■  be  a  -\-  c 

'J--    -^— , — ~j 1—  Aus.     , 

a.    +(10  —  a  r  —  l)  c  a  —  c 

^^     a;2-10r  +  21 


x^  -  11  .r  +  6  • 
By  §  106  the  (i.  C.  D,  is  found  to  be  x  -  3. 
(x2--10x+  21)  ~{x~  3)  X 


(a;3  -  11  X  +  6)  -f   (x  -  3)       t^  +  3  a-  -  2  ' 


Ans. 


X^-%X-?j  ^  x-3 


x^-Tar'^  +  l  ■■     x^^Sxi-l 

x'  -  10  .T  +  21  a; -3 

,.3  _  46./- -21'  ^'''-    x'^  +  7.r  +  3 


104  ALGEBRA. 


_     ^        ^^    ,    -  .  x^-^x-2 

36.      ,   .    ,    ., p.  Ans 


^.3_3^  +  2 

a;3  +  a.2  +  3^_5 

a;2-4a;  +  3       ' 

a;3_^2_7^_^3 

x'  +  2;r3^  2.x- 1* 

p 

«3^3^2^   +   3„J2^ 

\  +  2x'  +  x^^-2x* 

2  b' 

^'-        .^-4.  +  3      •  ^"^ 

,,;3    +    3^2_20 

.x^  —  ic^  —  12 

x^  —  ^"^  —  7  a;  +  3 

^^-    x^+2a^  +  2x-V  ^"^ 

o?  —  a^h  —  a  ly-  —  2h^ 
40.    ^^ ;— -r- — — :t-t-„  .  Ans 


a;^  +  5  a;  +  5 
a;2  +  2  a;  +  5 


a;  —  3 
a;2  +  5a;+  10 
a;3  +  2a-2  +  3x  +  6 
a;-3 


41.    .-    -  ^  :     '   "     '      "  .  .  Ans. 


a;2  +  1  • 
ft -26 
a  +  26" 
1  -  a;  +  2  x'-^ 
1  +  3a;2  +  2a-=*  +  3a;*'  """"     l-a^  +  Sar^" 

Case    II. 

117.    To  reduce  a  Fraction  to  an  Integral  or  Mixed  Number. 

1.    Reduce    — ~ —    to  an  integral  or  mixed  number, 

X  +  2 

a;+2).r'^  +  5a;-l(  ^  +  3  +  ^^ 
a;--^+2a;  or  t  " 

3a; -1  .x  +  3 ^    (§113) 

3  a; +  6  '-"  ^  ^ 

—  7 

Since  a  fraction  represents  the  quotient  of  the  numerator  divided  by 
the  denominator,  we  perform  the  indicated  division,  adding  to  the  quo- 
tient the  fraction  formed  by  placing  the  remainder  over  the  divisor. 
Hence, 

To  reduce  a  fraction  to  an  integral  or  mixed  mimber,  we 
have  the  following 

Rule. 

Divide  the  numerator  ly  the  denominator,  and  if  there  is  a 
remainder  place  it  over  the  divisor,  and  add  the  fraction  so 
formed  to  the  quotient. 


FRACTIONS.  105 


Reduce  the  following  to  integral  or  mixed  numbers  : 
^     a^  +  ab  +  c  ^     a'^  —  1 

9     1  +  '^" 


*'■ 

a 

3. 

a^^ab  +  b^ 

a 

4. 

4.ax-8bx-c 

2x 

5. 

a;2  +  4a;_3 

x  +  2        • 

6. 

a^-b^ 
a-b' 

12. 

a*  Jrtrb''  +  b* 

a^^ab^  b'  • 

13. 

a*  +  «^  +  1 
a^-l       ■ 

14. 

a;2  —  4  a  cc  +  3  a^ 

a^-2a 

15. 

a4  -  a^  4-  a'  +  1 

10. 

11. 


1  +  « 

X*  +  a;^  +  a;"^  +  1 
cc'-^  +  a;  +  1 

'a-b   ' 


Ans.     a^  +  2  +  -^ 


1 


Ans.     ft-  —  2  a  +  2  — 


a^  +  ft  +  1  a'^  +  ft  +  1 

Case    III. 
118.     To  reduce  a  Mixed  Number  to  a  Fractional  Form. 

1.    Reduce  a  -\-  b  -\ to  a  fractional  form. 

a  —  b 

b^  a^  _  h^  +  ^,2  ^2 

ft  +  ^^  + 


a  —  h  a  —  b  a  —  b 

2.    Reduce  x  —  if  -\ ~  to  a  fractional  form. 

x  +  y 

x-y  +  ""'  "^  ^'  =  ^'  -  /  +  (^  +  .V') 

_  a;^  —  I/'  -\-  x^  +  y^  _    2x- 
~  X  +  y  ~  X  -\-  y 


106  ALGEBRA. 

3.    Keduce  x  -\-  y —  to  a  fractional  form. 

x-y 

x^y-  '^'  +  ^  =  x'-f-  (.7-  +  f) 
x—y  x—y 

=  t-.t.-i:^'^-jt  =  ^^  ,    or  ^^ 
x  —  y  x—y'         y  —  X 

It  will  be  observed  by  referring  to  Case  II.,  of  which  this  is  the  con- 
verse, that  the  integral  part  of  the  mixed  expression  always  stands  for 
the  quotient,  the  denominator  of  the  fractional  part  for  the  divisor,  and 
the  numerator  for  the  remainder ;  and  that  the  dividing  line  also  per- 
forms the  office  of  a  vinculum  for  the  numerator.     Hence, 

To  reduce  a  mixed  number  to  a  fractional  form,  we  have 

the  following- 
Rule. 

Multiply  the  integral  part  by  the  denominator  of  the  fraetion  ; 
to  the  product  add  the  nuraerator  if  the  sign  of  the  fraction  is 
plus,  ami  subtract  it  if  the  sign  is  minus,  and  under  the  result 
write  the  denominator. 


Reduce  to  fractional  form  the  following  examples  : 

r 


+  '.  n.  A--X. 


5.  l+i.  12.    ^-1, 

X 

6.  1-1. 

X 

1 

7.  x-h-. 

y 

8.  .-^. 


y 


9.    ^-o. 


13. 

a  —  x  A . 

a  -\-  X 

14. 

2xy  +  y^ 

15. 

— .^V 

16. 

^.— • 

17. 

— ^- 

FRACTIONS.  107 


18.  ^:^_(«_,). 


19.    a^  —  2ax  +  4:  ./■- -—  .  Ans. 


a  +  2x  a  -\-  2x 

20.    ^^            +  ic  —  «  +  tj .  Ans.        -        ^-^-^ 

X  +  a  X  +  a 

91     1    .    ^'^  +  ^'  -  ^'                   ^^^  (a  +  b  +  c)  (a  +  b-  c) 

2ab         '                --•  2ab 

22.     1        g^  -  «^  -  1                    ^^,^_  (g  +  c  +  1)  (g  -  c  +  1) 


23.  ajs  _  ^2  _  .^  _  1  ^  J_^  ^  Aj,g_     ^l±i 

a;  —  1  cc  —  1 

24.  .^  +  .-U-l--:^^*.  An.    ,-iL 

0  —  o  0  —  o 


Case   IV. 

119.    To  reduce  Fractions  to  Equivalent  Fractions  having  their 
Least  Common  Denominator. 

1.    Reduce   ,   ,  and  —      to  equivalent  fractions  hav- 

7nx     my  mz 

ing  the  least  common  denominator. 

The  required  fractions  must  have  for  their  (lenominator.s  m  x  y  z,  the 
least  connnon  multiple  of  the  given  denominators. 

If  we  multiply  the  numerators,  a,  b,  c,  by  the  quotients  of  m  x  y  z 
divided  by  m x,  my,  m z,  respectively,  we  shall  have  the  e(]^iuvaleiit 
fractions  required. 

That  is,  -^  =:  .^^^AlA^  =  ^2^ 

m  X       mx  X  (y  ^)       "i  X  y  z 


b  b  X  (xz)  b 


xz 


ny        my  x  {xz)       mxyz 
c    _    c  X  (xy)    _    cxy 
nz       onz  X  {xy)       inxyz 


108  ALGEBRA. 

Hence  the  following 

Rule. 

Find  the  least  common  multiple  of  the  denominators  for  the 
least  common  denominator.  For  new  numerators,  midtiply 
each  numerator  hy  the  quotient  arising  from  dividing  this 
multiple  hy  its  denominator. 

Note  1.   Fractious  should  always  first  be  reduced  to  tlieir  lowest  terms. 

Note  2.  The  familiar  method  of  multiplying  together  the  denominators 
for  a  new  denominator,  and  each  numerator  by  all  the  denominators  except 
its  own,  for  a  new  numerator,  is  sometimes  useful.  When  the  denominators 
are  mutually  prime,  it  is  identical  with  the  process  above. 

Note  3.  Every  integral  form  may  be  considered  as  a  fraction  with  unity 
for  its  denominator  ;  that  is,  a  —  -  . 

2.    Reduce , -, ,    and  „    „       to  equivalent  fractions 

2xy     4:X  y-  ox'^  y 

having  the  least  common  denominator  (L.  C.  D.). 
The  L.  CD.  is  Sx^y\ 

a  b  ,        c 

2xy        Ax  y^  Sx^y 

are  equal,  respectively,  to 

Aaxy  2hx  .,        cy 

and 


8x^2/'^'       8x^3/2'  8x-\y2 

12  3 

3.    Reduce    -,   — j- ,   and  — — —j-  to    equivalent    frac- 

a'   a{a  —  b)  b{n  -\-  b) 

tion.s  having  the  least  common  denominator. 

The  L.  CD.  is    ab{a^-b'). 
1  2 


nd 


a '       a(a  —  b)'  b  {a  +  b) 
are  equal,  respectively,  to 

b  (g^  -  ft")           2b(a+J})  Sa(a-b) 

ab{a:'-¥)'       ab{a'-b')'  """'      ab(a''-h'') 


FRACTIONS.  109 

Reduce  to  equivalent  fractions  having  their  L.  C.  D.  : 

—  b      a  —  c      b  —  c 


4. 

a      b 
4'    6- 

5. 

2x     X      3x 

T'    6'    12* 

6. 

4a      3a 

5  b'    10  c' 

T. 

a            b 

3xy'    6xyz' 

c 

27 

8. 

",   ',    3x. 

y    ic 

9. 

be      ca 

11. 

12. 
13. 
14. 
15. 


ab    '      ac    ^      be    ' 
1  1 


a-b'    a?-V^ 
1  2 


x-2'    (x-2f 

1  ,2 

X  —  2/ '    x^  —  y'^' 

xy  X 

25  cc^  —  2/^ '    5x4-2/" 


16.    -^^  1 


x{x  +  y)      y{x  —  y) 

10   ^~^    ^-2    ^-'^      ij         y  ^- 

2      '        3      '        G      ■  '    x{x'-y')'    y  (x'  +  y') 

18        '   «  ^ 


19. 

20. 


4  (a  -  6)  '    8  (a^  -  62^  • 

5  6  2-3a; 

1  +  2x'    1  -  2a;'    1-40;^ 

1  2 


2a(a-b)'    3b(a^-b')'    Qc(a  +  b)' 

120.    Addition  and  Subtraction  of  Fractions 


1.    Find  the  sum  of        and   -, ,    and  also  their  difference. 

ac_adbc_ad  +  bc 
b'^  7l~Td'^  bl~        bd~ 

a       c  _ad       be       ad  —  be 

b~d~Vd~V7i^      hd 


110  ALGEBRA. 

Hence,  for  adding  or  subtracting  fractions,  we  have  the 
following 

Kule. 

Reduce  the  fractions,  if  necessary,  to  equivalent  fractions 
having  tJieir  least  common  denominator ;  then  add  or  subtract 
their  numerators,  as  the  sign  before  the  fraction  directs,  and 
write  the  restdt  over  the  least  common  denominator. 

2.  Simplify    — j +     ., . 

4a3  2x 

TheL.  C.  D.  is  Ax. 

x  —  3      X  +  3  _x  -3  +  2{x  +  3) 
4x  2x  4x 

_a;-3  +  2a;  +  6_3(a:+l) 
~  4cc  ~       4x 

3.  Simplify   ''"•''*       ^"^ 


4        ' 

3 

rhe  L.  C.  D.  is  12. 

x  —  3      x  —  4: 
4       '       3 

3(x 

-3)  +  4(:r-4) 
12 

3x- 

-9  +  4a;-lG      7a;- 25 

12                          12 

Simplify 

->'^- 

i 

8.    A  +  J-, 
ax       ox 

^    x+3      x-6 
^•2              3     • 

9. "+"-:. 

X       x^ 

a; -3      x-5 
^•6              4      • 

ar-a       x-b 

^^-        a               b      ' 

a       b 

12        1 

x^       ax       a* 

FRACTIONS.  Ill 

13.  Simplify  "^*  +  ?±-*. 

a  +  b       a  —  b 

a  +  h     a  —  b  a^  —  b^ 

_  a^  -  2  a  b  +  b^  +  a^  +  2  ab  +  b^  _  2a^  +  2  b^ 

a^-b^                       -  a^-h^ 
Simplify 

14.  -'.+^.  21.  °      .+      1 


a+6       a  —  6  "x(a  —  x)       a  —  x 


a  X 


15.  — ^ ^.  22.       ,  ,  , 

X  —  y      X  -{■  ij  X  (a  —  X)       a  {a  —  x) 

16.  -^+-^-.  23.       "  ^ 


a;  +  a       x  —  a'  '    x  —  1       (x  —  1)'^' 

17.    .^^^ ^.  24.     A^  +  -J^+      2^ 


x  -\-  a       X  —  a  a  -\-  h       a  —  b       a?  —  h^ 


2  1 


18.    ^  -  -  .  25.    — —  +  ^^  +  1. 

X  —  1       X  a  +  X       a  —  X 

19.^+1-^.  26.^  +  -,"^,. 
4y          X  -\-  y  a  —  X       x^  —  a^ 

20.    ^  +  -1^.  27.    ^L+      1  2* 


121.    Multiplication  and  Division  of  Fractions. 

1.   Multiply  ?  by  ^ . 

Let 


then 
and 


But 


-^  =  x,      and      ^  = 

y 

a  =  bx,  and      c  = 

dy 

ac  =  bdxy 

(Ax.  3) 

ac 

bd  =  ''y 

(Ax.  4) 

a        c 
''-■=b><d 

a       c  _ac 
b  ^  d~  bd 

112  ALGEBRA. 

Hence,  for  the  multiplication  of  fractions,  we  have  the 
following 

Rule. 

Multiply  the  nuinerators  together  for  a  new  numerator,  and 
the  denominators  for  a  new  denominator. 

2.  Divide  ?  by  ^. 

0         d 

Let  T  =  a^j     and      ,  =  y 

b  d       ^ 

Then  a  =  bx,    and     c  =  dy 

ad  —  bdx,  and  be—  bdy  (Ax.  3) 

ad  _  bdx  _  X 

be        b  dy      y 

x>  i.  X  a        c 

But  -  -x-^  y=  T  ~  -J 

y  "      b       d 

a    .    c  _  ad 

b    '   d  ~  b  c 

Hence,  for  the  division  of  fractions,  we  have  the  following 

Rule. 

Invert  the  divisor,  and  then  proceed  as  in  multiplication  of  a 
fraction  by  a  fraction. 

Note  1.  As  a  mixed  number  can  be  reduced  to  a  fractional  form,  §  119, 
and  an  integral  expression  can  be  expressed  as  a  fraction  by  writing  under  it 
1  as  its  denominator,  these  rules  cover  all  possible  cases  in  multiplication  and 
division  of  fractions. 

Note  2.    Alwai/s  cancel,  if  possible. 

6  a^  14  x^ 

3.  Multiply   35^3  by  -g^. 

6a^        Ux 
35x3  X   9^3 

4.  Multiply     "7    , 

a^  -\-  ab 

a-b  a^-b^  _ 

a^  -\-  ab       a^  —  ab 


2      3-2 

"7-5 

.  2 
•3 

■  la^x^ 

■  3a8a;«~ 

4  a 
Wx 

^^^ 

ab 

a-b 

n  (r,   JL    h\ 

X 

{a  +  b)  (a 

-b) 
7.^ 

a 

-b 

FRACTIONS.  113 

o.    Divide     _    .,      by  z^ . 

ox-//         loxy 

AaH   ^    2_ab^  _  4  aH       loxif  _  2  -2  •  3  •  oa-bxy'  _  Gay 
^x^y   '    15 xy'^       ?>xry        'lab-  o-2ab^x^y        ^    hx 

r.     T^-   •  1       «"  —  4  X'    ,      a^  —  2  a  ic 
6.    Divide    —, : by 


a^  -\-  ^ax     "    ax  +  4:X^' 

(V-  —  ^3?        a-  —  2  ax        a^  —  4  ce"^        a  a;  +  4  cc^ 
.J ^ Y^  

a^  +  4aa;   "    ax  -{■  ^x^       a^  +  4ax       c?  —  2  ax 

(a  +  2a;)  (a  — 2x)       ic(a  +  4a;)       a;(a  +  2a;) 


«  (a  +  4  cc)              a  (a  — 

2x) 

a^ 

Simplify 

^     4x3         a 
a^        2  x^ 

12. 

x^            a;  +  y 

(a^  +  2/)-^''      y      ' 

2>x-       by-        12  xz" 
^-     4.Z   '^  ioxy''^  20x^f-- 

13. 

tt                    a 

x^  —  a^  '   x(x  —  a)' 

^     4x3   .   2a 
a-     '     X  ' 

14. 

x^      .      y 

(x  +  ?/)^  ■   X  +  2/  ■ 

6a^    .    9a3 
14x3  •  35^2- 

15. 

^2  _^   3,2           ,^3   ^   ^3 

a2  _  ^2  X  «4  _  ^4  • 

11        ^  +  3     ^     ^^ 

16. 

x^-y^             x-y 

X3   +   y3     •     a;2  _  3,  ^   _^   y2  • 

a                a           a'^-b'' 

1<.     J  X  7  X  j — 

a  +  b       a  —  b          ab 

18.  ^^^-'"x  !7.r 

x^  —  2o           X-  +  2  X  - 

9 

x  +  1 
— — ^^  .                Ans.    X, 
x'^  +  0  X 

^^     ab  +  b'   .   a'x  +  b'x 
n^  —  a6    '     a^c  —  b^ c  ' 

bc(a  +  b) 

«  X  (a^  —  a  6  +  o"^) 

20.    a  +  lU(l  +  lV 

x2  -  X  +  1 
Ans.     s • 

114  ALGEBRA, 


CHAPTEK    XV. 
GENERALIZATION. 

122.  Since  letters  stand  for  any  numbers  whatever,  the 
answer  to  a  problem  in  which  the  given  numbers  are  repre- 
sented by  letters  is  a  general  expression  including  all  cases  of 
the  same  kind.  Moreover,  the  operations  performed  with 
letters  are  not,  as  with  figures,  lost  in  the  combinations  of 
the  process,  but  all  appear  in  the  resulting  expression.  Such 
an  expression  is  called  a  formula,  and  when  expressed  in 
words,  a  ride. 

123.  To  illustrate  the  making  of  formulas,  and  their  use, 
the  following  problems  are  solved. 


1.   Divide  a 

into  two  parts  such  that  the  greater  exceeds  the 

less  by  b. 

Let 

a:  =  tlie  greater  part  ; 

then 

a;  -  6  =  the  less  part. 

.«. 

%x-h=a 

2x=a+h 

X  =  — —  ,  the  greater  part ; 

x-b=  — H— >  the  less  part. 

In  this  example  a  represents  the  sum  and  b  the  difference  of  two 

numbers,  and  ^ — - — ,  — -— ,are  respectively  the  formulas  for  the  two 

numbers.  Stated  in  words,  the  formula  is  as  follows  :  AVhen  the  sum 
and  difference  of  two  numbers  are  given,  to  find  the  numbers  we  have 
the  following 


GENERALIZATION.  115 

Rule. 

(1)  Divide  the  mm  plus  the  difference  of  the  two  niombers  hy 
two,  and  it  will  give  the  greater. 

(2)  Divide  the  sum  minus  the  difference  of  the  two  nwmhers 
bg  two,  and  it  will  give  the  less. 

Find  the  numbers  when 

2.  The  sum  is  14  and  the  dili'erence  is  6. 

3.  The  sum  is  25  and  the  difference  is  7. 

4.  The  sum  is  50  and  the  difference  is  8. 

5.  The  sum  is  100  and  the  difference  is  20. 

6.  The  sum  is  18  and  the  difference  is  12. 

7.  The  sum  is  29  and  the  difference  is  9. 

8.  The  sum  is  105  and  the  difference  is  15. 

9.  The  sum  is  53  and  the  difference  is  7. 


10.    The  sum  is  2  x  and  the  difference  i 


9., 


11.  The  sum  is  6x^  —  '3g  and  the  difference  is  2  ic^  +  y. 

12.  The  sum  is  3a  —  6c  and  the  difference  is  5  a.  +  4 1^. 

13.  Two  boys  20  rods  apart  walked  directly  toward  each 
other  till  they  met.  If  one  walked  4  rods  more  than  the  other, 
how  many  rods  did  each  walk  ? 

14.  If  A  can  do  a  piece  of  work  in  a  hours  and  B  can  do  it 
in  b  hours,  how  long  will  it  take  A  and  B  together  to  do  the 
work  ? 

Let  X  =  the  reciuired  number  of  hours 

4.1,  1,11 

then  -  +  -  =  - 

a      0      X 

bx  +  ax  =  ab 
a  b 

a  +  b' 

That  is,  — —7  is  the  formula  that  expresses,  in  terms  of  the  time  in 
which  each  can  do  it  alone,  the  time  it  will  take  two  men  together  to  do 
a  given  piece  of  work.     Stated  in  words,  the  formula  is  as  follows  :  When 


116  ALGEBRA. 

the  time  it  takes  each  of  two  men  to  do  a  piece  of  work  is  given,  to  find 
the  time  it  will  take  the  two  to  do  it  together,  we  have  the  following 


Divide  the  product  hy  the  sum  of  the  numbers  expressing  the 
time  it  will  take  each  to  do  it  alone. 

15.  If  Charles  can  do  a  piece  of  work  in  3  days  and  James  in 
2  days,  how  long  will  it  take  both  together  to  do  it  ? 

Ans.     ;^^  ,  or  11  days. 

16.  If  of  two  pipes,  one  will  fill  a  cistern  in  4  hours  and  the 
other  in  5  hours,  how  many  hours  will  it  take  both  running 
together  to  fill  it  ? 

17.  If  A  can  build  a  certain  piece  of  wall  in  7  days  and  B  in 
10  days,  how  many  days  will  it  take  A  and  B  together  to  build 
the  wall  ? 

SIMPLE    INTEREST. 

124.  Though  the  first  letters  of  the  alphabet  usually  repre- 
sent known  numbers,  and  the  last  unknown  numbers,  it  is 
often  convenient  in  formulas  to  use  the  initial  letters  of  the 
names  of  quantities  whether  known  or  unknown. 

In  the  following  examples  in  interest  let  p  represent  the 
principal,  r  the  rate,  t  the  time,  i  the  interest,  and  a  the 
amount.  It  must  be  understood  that  t  represents  the  time  in 
years,  and  that  r  is  a  decimal  whose  denominator  is  100,  and 
represents  the  interest  of  a  unit  of  the  principal  for  a  year. 

18.  What  is  the  interest  of  ^j  dollars  for  t  years  at  r  per  cent  ? 
What  is  the  amount  ? 

The  interest  =  principal  X  time  X  rate, 
or  i—x>tr  (1) 

The  amount  =  principal  +  interest, 
or  a  =  p  +  p^'''  (2) 


GENERALIZATION.  117 

These  formulas  contain  four  different  things,  of  which  any  one  may 
be  determined  when  the  others  are  known. 

Thus  from  (I)  and  (2),  when  a,  p,  and  r  are  known,  we  have 

i  a  ~  p 

t  =  — ,  or  ~ 

2)  r  'p  r 

Hence  to  find  the  time  when  the  amount,  principal,  and  rate  are 
known  we  have  the  followins 


Rule. 

From  the  amoimt  subtract  the  principal,  and   divide   the 
remainder  by  the  product  of  the  principal  and  the  rate. 

19.  From  equation  (1)  find  the  formula  for  the  principal  when 
the  interest,  time,  and  rate  are  given. 

State  the  rule. 

20.  From  equation  (2)  find  the  formula  for  the  principal  when 
the  amount,  time,  and  rate  are  given. 

State  the  rule. 

21.  From  equation  (1)  find  the  formula  for  the  rate  when  the 
principal,  time,  and  interest  are  given. 

State  the  rule. 

22.  From  equation  (2)  find  the  formula  for  the  rate  when  the 
principal,  time,  and  interest  are  given. 

State  the  rule. 

23.  How  long  must  $200  be  on  interest  at  6  %  to  gain  $36  ? 

24.  How  long  must  $500  be  on  interest  at  4  %  to  amount  to 
$620? 

25.  At  what  rate  must  $300  be  put  on  interest  to  gain  $63 
in  3  years  ? 

26.  What  principal  at  5  %  will  gain  $400  in  8  years  ? 

27.  Ho.v  long  will  it  take  a  sum  of  money  on  interest  at  6  % 
to  double  itself  ?     At  5  %  ?     At  4  %  ? 


118  ALGEBRA. 

PRESENT    WORTH    AND    DISCOUNT. 

125.  The  Present  Wortli  of  a  debt,  payable  at  a  future  time 
without  interest,  is  a  sum  of  money  which,  put  at  interest, 
will  amount  to  the  debt  at  the  time  of  its  becoming  due. 

The  debt,  then,  is  an  amount,  the  'p'>'^scnt  worth  is  the  prin- 
cipal, that  is,  we  have  the  amount,  time,  and  rate  given  to 
find  the  principal. 

In  formula  (2),  §  188,  j;  stands  for  the  present  worth. 
f  +  p  tr  =  a 
(l  +  tr)]p  —  a 


'-        1  +  <  r 
Hence  to  find  the  present  worth  of  an  amount  when  the  time  and  rate 
are  given,  we  have  the  following 

Kule. 

Divide,  the  amount  hy  one  plus  the  prodicd  of  the  time  and 
the  rate. 

28.  What  is  the  present  worth  uf  loOO  due  in  4  years  at  5  %  ? 

29.  What  is  the  present  worth  of  $560  due  in  2  years  at  6  %  ? 

30.  What  is  the  present  worth  of  $440  due  in  2  years  at  5  %  ? 

31.  What  is   the  present  worth  of  $1000  due  in  3  months 

at  6  %? 

The  Discount  is  the  interest  on  the  present  worth,  and  is 
equal  to  the  amount  due  minus  its  present  worth. 

32.  Find  the  discount  on  $336  due  in  3  years  at  4  %. 

33.  What  is  the  discount  on  $672  due  in  2  years  at  6  %  ? 

34.  What  is  the  discount  on  $220  due  in  4  months  at  6  %  ? 

35.  What  is  the  discount  on  $500  due  in  3  months  at  4  %  ? 

36.  What  is  the  discount  on  $1000  due  in  2  months  at  6  ^  ? 


MISCELLANEOUS  EXAMPLES.  119 


CHAPTER    XVI 
MISCELLANEOUS    EXAMPLES. 

Solve  the  equations  : 

1.  2  a;  +  3  =  IG  -  (2  X  -  .3). 

2.  :r  -  (4  -  2  x)  =  7  (x  -  1)  . 

3.  5(4-3^;)  =  7(3-4x). 

4.  2  {x  -  3)  =  5  (x  +  1)  +  2  a-  -  1. 

5.  4  (1  -  r;  +  3  (2  +  x)  =  13. 
G.  5  (a;  +  2)  =  3  {x  +  3)  i-  1. 

7.  8  (9  -  2  a-)  -  17  (25  -  3  x)  =  -  3. 

8.  7  (8  -  3  a-)  +  G  (2  X  -  o)  =  -  28. 

9.  3  a;  -  4  {9  -  (2  a;  +  7)  +  3  a-}  =  13. 

10.  7  X  -  {4a-  -  1  -  (Ga;  +  4)}  r=  27  -  2  [5.r  -  (3a;  +  2)]. 

11.  (2  X  -  3f  -  (2  a-  -  7)-  =  5  (a-  +  3> 

12.  (7  +  a-)  (2  -  a;)  =  (2  -  x)  (5  -  a;)  -  (2  a:  -  5)  (a-  -  5). 

13.  2  a-  -  [3  -  {ix  +  (x-  1)}  -  5]  =  8. 

14.  ^'+"^  +  ^  =  10. 

J  .3  4 

17.    l(2x  +  5)^l(3x-S)=:r}^(4x-3). 

._     3a;-8       2a:  +  o       5a;-G_4a;  +  3       14 

^^-  ~9  3~  +  ~7r~-T5~~¥- 


120  ALGEBRA. 

Sx  .  2x-4:      5x      22 


19. 


5^3  3         3  ' 


90    x  +  7      x-7  _x      1 
'      14  7      ~S^8' 


a;  —  12_5a:      x  -\-  3 
^5       ~  T  3~ 


26.  4.-^%6=2^^f^-l. 

3  4  4 

„^7x  +  5       6a;  —  30 

27.  — -^  +  -^- ^  =  x+l. 

7  7a;  —  ( 

2g    284  -  4 a-      75  -  3 a;  _  22-2: 


12 

oo    17       3  + 5a;       .          3 a; +  3       18 -5 a; 
J9.   17 ^-4..  =  -^ 3—. 

30.   ?^  +  3^+10  =  3.-^-^- 

.        ^  _  3  (x  +  1)  _  4  (a;  -  7)  _  7  a:  -  17 
"  ■    21  11        ~         7  10       • 

4       8  —  5  a;       5  a?       10  a;  —  7       „ 

'^''-   3 2  4   =—3 ^• 

33.   6  -  2  (4  a>  -  1)  =  J  (2  -  5  aO  -  H3  iP  +  2). 

■•^•--(I-|)  =  -'- 

or    o  ^  /•'■   ,   ^\       rr       1  /a;      a;\ 


MISCELLANEOUS   EXAMPLES.  121 

36.  ^x-\-^(2-x)  =  \{2x-^  (5  +  X)}  -  H^  -  5). 

37.  0.3  X  -  5  +  0.65  x  :=  0.5  a;  +  8.5. 
Transposing  and  uniting,     0.45  x  =  13.5 
Dividing  by  0.45,  cc  =  30 

0.75 +  :r  0.25  -  X 

^^-   ~0j2r  =  ^^-      0.25      • 
Performing  the  indicated  division,    6  +  8  aj  =  15  —  1  +  4  x. 

39.  1.1  X  -  0.25  =  0.75  x  +  0.8. 

40.  0.5  x  —  0.6  X  +  0.65  x  —  0.775  x  +  15  =  0. 

41.  2.5 x  —  \=  0.25 X  +  2x  +  0.2 x. 

42.  1.2  X  +  0.05  r=:  0.07  x  +  0.3  a;  +  16.65. 

43.  0.75  X  -  0.375  +  2  =  x-  0.25  +  0.125  x. 

44.  2.4. -°-^'^— "■"^  =  0.8. +  8.9. 

0.5 

,.    n-.,.         0.135^-0.225      0.36      0.09  x- 0.18 
4o.    0.15.  + =  _ ^^_. 

4:&.  a  +  X  —  h  =  a  -{■  b.  A'^.    a  x  ~  b  —  a  —  b  x. 

4:1,  x  —  2b  —  2a  —  X.  AQ.   a  X  —  a^  =  b  X  —  b^. 

50.  ax  —  a  —  6=    c  —  b  x  —  ex. 

51  ax  —  bx  —  a'^  +  b-  =  0. 

02.     -  _  6  r=  -  +  -  +   fi. 

a  r        d 

Clearing  of  fractions  and  transposing, 

c  d  X  ~  a  d  x  —  a  c  X  ^=  a  b  c  d  -{-  a  c  d  e 

(c  d  —  ad—  a  c)  X  ^^  ab  f  d  -\-  a  cd  e 

abcd-\-  a  c  d  e 

cd  —  ad  —  ac 

53.  <t  {x  —  b)  -\-  a  X  —  b  =  b  X  —  a. 

54.  a  {x  -  2h)  -^  b  {x  -  2  c)  ^  e  {x  -  2  a)  =  a^  ^  b''  ^  c\ 

55.  -4-7  =  1.  56. a^  —  b- .  • 

ah  b  a 


122  ALGEBRA. 


^„     ax       hx       ,,         „       ax       bb  -{-  ax 

^^-  T-T  +  ^'  ="-T 6— • 

58.  'I  (a;  -  2  a)  +  ^  (a-  +  ^.)  =  1. 

bx+  1       a  (x'  -  1) 

59.  ax =  — ^ . 


X 


—  X       2  X      a 


61. 


bx  .     b  b         X 

ab  —  x^       4:X  —  a  c 


ex 


62.    am-b-'^+'^  =  0. 
b        m 

Sax  — 2b       ax  —  a_ax       2 

3b  2b~  ^~b'~3' 

b  -\-  X       b^  —  X       X  —  b       ab  —  x 


04. 


b'  aH  «2  p       ' 

65.  One  flock  of  sheep  consists  of  two  slieei)  more  than  half 
of  another  flock.  They  both  together  amount  t<.  101  sheep.  How 
many  are  there  in  each  ? 

66.  John  has  80  cents  and  James  has  15.  How  many  cents 
must  John  give  to  James  in  order  that  he  may  have  just  four 
times  as  many  as  James  ? 

67.  A  can  do  a  piece  of  work  in  15  hours,  hut  witli  tlie  helji 
of  B  lie  gets  it  done  in  5  hours.  In  what  time  can  B  do  it 
alone  ? 

68.  A  spends  .\  of  his  income  in  board,  ^  in  clothes,  y'^  in 
sundries,  and  has  $212  left.     Wliat  is  his  income  ? 

69.  f  of  A's  money  is  ecpuil  to  I','s,  and  7j  of  B's  is  equal  to 
C's.     In  all  they  have  1770.      How  nmch  has  each  ? 

70.  What  is  the  property  of  a  ])erson  whose  income  is  $430, 
when  he  has  ^  of  it  invested  at  4  %,  \  a,t3  7o,  and  the  remain- 
der at  2/.?  Ans.     $12000. 


MISCELLAXKOUS   EXAMPLES.  llio 

71.  A  testator  left  I  of  his  estate  to  his  widow,  -J-  to  eacli  of 
his  two  sons,  -^  to  his  servant,  and  the  residue,  |G00,  to  chari- 
ties.    What  was  his  whole  estate  ? 

72.  A,  P.,  C,  and  D  divide  12520  as  follows  :  C  has  $360,  B 
as  much  as  C  and  D  together,  and  A  $1000  less  than  twice  as 
much  as  B.     What  is  the  share  of  each  ? 

73.  In  a  mixture  of  oats  and  harle_y,  the  oats  are  25  bushels 
more  than  half  of  the  mixture,  and  the  barley  5  bushels  less  thau 
a  third  of  it.      Plow  many  bushels  are  there  of  each  ".' 

74.  A  starts  from  a  certain  place  and  travels  at  the  rate  of  7 
miles  in  3  hours  ;  B  starts  from  the  same  place  6  hours  after  A, 
and  travels  in  the  same  direction  at  the  rate  of  5  miles  in  2 
hours.     How  far  will  A  travel  before  he  is  overtaken  by  B  ? 

75.  A  market-woman  bought  a  certain  number  of  eggs  at  the 
rate  of  5  for  2  cents,  and  suld  half  of  tliem  at  2  for  a  cent,  and 
half  of  them  at  3  for  a  cent,  and  gained  four  cents.  How  many 
eggs  did  she  buy  '.'  Ans.    240. 

76.  An  officer  can  form  the  men  of  his  regiment  into  a  hollow 
square  6  dee[).  ,The  number  of  men  in  the  regiment  is  1320. 
Find  the  "number  of  nien  in  the  front  of  the  hollow  square. 

Ans.    61. 

77.  An  officer  can  form  his  men  into  a  hollow  s(|uare  6  deep, 
and  also  into  a  solid  square  36  in  front.  Find  the  number  of 
men  in  the  front  of  the  hollow  square.  Ans.     60. 

78.  A  cistern  is  tilled  by  either  one  of  its  three  i:)ipes,  running 
alone,  in  f,  h,  and  r  minutes  resjiectively.  How  long  will  it 
take  the  pipes  to  fill  it  when  they  are  all  running  together  ? 

X  —  6  X  —  b 

80.    ^  +  ^-Llll  =  2. 

X  —   1  X 


81. 


3  _      2      _         5 

X       a:  +  1        4  (,r  +  1) 


124 

ALGEBRA. 

82. 

3           1             12 

2x--5      x~x(2x-5)' 

83. 

6             4x             3 

X  —  1       a;'^  —  1  ~  a;  +  1  ■ 

84. 

7           6  X  +  1       3  +  Gx^ 
a;-l~a;  +  l     '     1  -  x-  ' 

85. 

4          7              37 

X         X  +  1         X'  +  x' 

86 

4         7    _        sr 

X  +  2   '   X  +  3       ./•■-  +  T)  x  +  G' 

87 

4                11 

ai'-l'l  +  x       1—x' 

88 

1                 3        _             5 

X  —  o   '    2  a;  -  (5       .r'^  —  8  .r  +  15 

89. 

x  +  aa^  +  i       ^ 
X  —  b       X  —  a 

90. 

1                1              ./  -  /> 
X  —  a       X  —  b''  X-  —  ab' 

91. 

x  +  n       x-b       2(a  +  h) 
X  -~<i       X  ^  b~          X 

92. 

(5x-3yr=-5.                           f 
'l2x-r>>/  =  -4().                 .       j 

.T   +    -^ 


.,      (8.r-2l7/  =  5.  ■*'*■      I  .,     _2_26 

•  "•     iGx  -\-  14  7/  =  -  20.  i  "  "^       y  "  3  ' 

I  4         .3  [  y       X 


[•!  +  5,/  =  -4. 
95.     <  ''  98. 


'  +  '  =  ? 

a?        ?/  0 

1  _  1  _  1 

X       y  6 


MISCELLANEOUS   EXAMPLES. 


125 


99. 


100. 


101. 


102. 


103. 


101. 


U     y 

<  X  +  //  =  a. 

'(x-y=b. 

(  a  X  -\-  h  y  — 

c. 

(  m  X  -\-  n  y  = 

=  d. 

1  a       h 

[b       a 

(a       h 

,  r^ +  .  =  '"• 

*  +  "  =  „. 

^'ix-2y^ 
.  3cc  +  2y  + 
(  5  X  -  2  //  + 

?,z 

2z 

2z 

105. 


106. 


101 


108. 


109. 


(2x 
}3x 
(2x- 


-  3  y  +  4  «  =  9. 
-by-2z^-4.. 
Q>y-oz  =  -lS. 


13. 


^  X  +  y  =  21. 

hy  +  z^-2. 
(  a:  +  -  =  —  i. 

^x  +  y  =  a. 
-  y  ^z^h, 
i  x  +  z  =  c. 

^  a  x  +  b  y  —  1. 

hy  +  cz=^l. 
\ax  -\-  c  z  —  1. 

X       y 

y       z 

1        1 

+  -    =1. 
X        z 


110.  Two  brothers  received  by  will  equal  sums  of  money. 
After  several  years  the  elder  had  increased  his  by  25  %,  while 
the  younger  had  lost  50  %  of  his.  Then  the  elder  had  $2400 
more  than  the  yoiinger.      What  sum  did  the  younger  have  ? 

111.  A  workman  agrees  to  work  at  $2.50  a  day,  and  to  for- 
feit $3  for  every  day  he  does  not  work.  At  the  end  of  33  days 
he  receives  $44.      How  many  days  did  he  work  ? 


112.    How  many  days  must  the  workman 
from  his  work  to  receive  nothins:  ? 


Ex.  Ill  be 


113.  A  railroad  train  ran  a  certain  distance.  If  the  rate  had 
been  f  of  its  actual  rate,  the  time  would  have  been  21\  hours  ; 
and  if  the  rate  had  been  24  kilometers  more  an  hour,  the  train 
would  have  completed  f  of  the  distance  in  8  hours.  Find  the 
distance.  Ans.     924  kilometers. 


126  ALGEElixV. 

114.  A  man  paid  $1725  with  gold  eagles  and  silver  dollars. 
There  were  54  eagles  as  often  as  there  were  35  silver  pieces. 
How  many  coins  of  each  kind  were  there? 

115.  A  farmer  paid  $1600  for  24  acres  of  land.  One  part 
gave  him  a  revenue  of  4^-  %,  and  the  other  of  oi  %,  and  the 
total  revenue  was  163.      Find  the  area  of  the  parts. 

116.  A  man  wills  his  property-  to  his  children,  in  such  a  way 
that  the  eldest  is  to  have  $500  and  an  eighth  of  the  remainder, 
the  second  SIOOO  and  an  eighth  of  the  second  remainder,  the 
third  $1500  and  an  eighth  of  the  third  remainder,  and  so  on  to 
the  youngest.  The  legacies  are  thus  equal  to  one  another. 
Find  the  value  of  the  property  and  the  numher  of  children. 

Ans.     $24500  ;  7  children. 

117.  A  merchant  sold  a  piece  of  cloth  so  as  to  make  12  %  of 
the  sale.  The  prolit  is  1200  more  than  -^^^  of  the  price  paid. 
Wliat  was  the  price  paid  ? 

118.  The  following  is  said  to  he  inscribed  on  the  tombstone 
of  Diophantus  of  Alexandria  :  ''  One  sixth  of  his  life  he  spent  in 
boyhood ;  one  twelfth  in  youth  ;  he  was  then  married,  and  passed 
5  years  more  than  one  seventh  of  his  life  with  his  wife  before 
having  a  son,  whom  he  survived  4  years,  and  who,  dying,  was 
one  half  the  age  of  the  father."  How  old  was  Diophantus  at 
his  death  ? 

119.  There  is  a  certain  number  of  two  figures  whose  sum  is 
10.  Tf  the  order  of  the  figures  is  reversed,  the  number  thus 
formed  is  .'54  more  than  three  times  the  first  number.  Wliat  is 
the  number  ? 

120.  A  vase  full  of  water  weighs  12065  grams  ;  full  of  oil, 
11785  grams.  The  oil  weighs  0.912  as  much  as  the  water. 
What  is  the  weight  of  the  vase  ?  Ans.     8883/^  grams. 

121.  },  of  the  value  of  a  ]iiece  of  silk  is  equal  to  J  of  the 
value  of  a  ])iece  of  woollen.  Tlie  difference  of  the  two  values  is 
$38.40.  The  lengtli  of  the  woollen  is  ^  ^^  ^^^^  length  of  the 
silk,  and  a  yard  of  the  woollen  is  worth  $1.60.  What  is  the 
lengtli  of  each  ?  Ans.     Silk,  18  yards  ;   woollen,  60  j-ards. 


MISCELLANEOUS    EXAMPLES.  127 

122.  A  can  full  of  milk  weighs  2520  grains,  and  full  of  oil, 
2440.  The  milk  weighs  0.95  as  much  as  pure  water,  and.  the 
oil  0.9.      Find  the  weight  of  the  can.  Ans.     1000  grams. 

123.  I  am  twice  as  old  as  you  were  when  I  was  as  old  as  you 
now  are.  When  you  are  as  old  as  I  am  now,  the  sum  of  our 
ages  will  be  117  years.     What  is  m3^  age  ? 

124.  A  man  placed  at  interest  15068. <S0  at  5  %,  and  7  months 
later  ^4928  at  6  fo.  When  will  the  interest  on  the  two  sums 
be  equal  ? 

125.  There  are  three  casks.  The  second  contains  |  as  much 
as  the  first,  and  the  third  J  as  much  as  the  second,  and  50  liters 
less  than  the  first.     Find  the  caj^acity  of  each  cask. 

126.  A  poulterer  sold  all  his  eggs  to  4  persons  :  to  the  first 
4  of  all  he  had  plus  4  of  an  egg  ;  to  the  second  4  of  the  remain- 
der plus  4  of  an  egg  ;  and  so  on  to  the  rest,  to  each  4  of  the 
remainder  plus  4  of  an  egg.  No  egg  was  broken.  Find  the 
number  of  eggs  sold,  and  the  number  to  each. 

Ans.    624  eggs.     To  1st,  500 ;  2d,  100  ; 
3d,  20  ;  4th,  4. 

127.  From  Paris  to  Lyons  is  512  kilometers.  A  train  starts, 
from  Paris  for  Lyons  at  8  h.  45  m.  A.  M.,  at  the  rate  of  30  kilo- 
meters an  hour  ;  another  train  starts  from  Lyons  for  Paris  at 
Ih.  15  m.  p.  M.,  at  the  rate  of  28  kilometers  an  hour.  How  far 
from  Lyons  and  at  what  time  will  the  trains  pass  each  other  ? 

Ans.    182  kilometers;  7  h.  45  m.  p.m. 

128.  A  farmer  emplo3's  a  man  and  a  boy.  To  the  man  he 
pays  twice  as  much  daily  wages  as  to  the  boy.  For  15  days' 
work  he  gives  the  man  |37.50  and  10  gallons  of  maple  syrup, 
and  to  the  boy  for  12  days'  work,  $16.50  and  2  gallons  of  maple 
syrup.     What  is  the  price  of  the  syrup  a  gallon  ?     Ans.    $0.75. 

129.  For  15  pounds  of  coffee  and  12  pounds  of  sugar  one  pays 
$6.21  ;  and  for  17  pounds  of  coffee  and  14  pounds  of  sugar, 
$7.07.      Find  tlie  price  of  each  a  pound. 


128  ALGEBRA. 

130.  A  certain  capital  is  put  at  interest  for  a  year.  If  the 
rate  were  1  %  more  and  the  capital  $200  more,  the  interest 
would  be  $16  more  ;  if  the  rate  were  2  %  more  and  the  capital 
$300  more,  the  interest  would  be  $30  more.  Find  the  rate  and 
the  capital.  Ans.     Rate,  4  %  ;  capital,  $600. 

131.  Find  two  numbers  such  that  a  third  of  the  first  exceeds 
by  \  a  fourth  of  the  second,  and  four  thirds  of  the  first  minus 
two  fifths  of  the  second  is  equal  to  three  fourths  of  the  first  jjlus 
nineteen  fortieths  of  the  second. 

132.  Two  barrels  full  of  oil  that  is  worth  17  cents  a  quart 
are  sold  for  prices  that  differ  by  $4.08  ;  and  f  of  the  capacity 
of  one  barrel  is  equal  to  \%  of  the  capacity  of  the  other.  Find 
the  capacity  of  each  barrel.        Ans.     288  quarts  and  312  quarts. 

133.  Two  masses  of  iron  are  such  that  f  of  the  first  weighs 
96  pounds  less  than  |  of  the  second,  and  f  of  the  second  as  much 
as  %  of  the  first.     Find  the  weight  of  each. 

134.  Two  boys  Avork  together,  and  the  wages  of  the  first  are 
I  the  wages  of  the  second.  The  first,  who  works  5  days  more 
than  the  second,  receives  |20,  and  the  second  $12.  What  are 
the  daily  wages  and  the  number  of  days  for  each  ? 

135.  Two  cloaks  are  made  for  two  sisters.  For  the  elder  it 
takes  12  yards  of  woollen  and  8  yards  of  silk  lining,  and  for  the 
younger  6  yards  of  woollen  and  5  yards  of  silk  lining.  The 
cloak  for  the  elder  cost  .$7.96,  and  the  cloak  for  the  younger 
$4.15.  What  was  the  price  for  a  yard  of  woollen  and  for  a  yard 
of  silk  ? 

136.  If  the  pages  of  this  book  had  an  average  of  3  lines 
more  on  a  page  and  4  letters  more  in  a  line,  they  would  have 
224  more  letters  on  a  page  ;  but  if  they  had  2  lines  less  on  a 
page  and  3  letters  less  in  a  line,  they  would  have  145  letters 
less  on  a  page.  How  many  lines  are  there  on  a  jiage,  and  how 
many  letters  in  a  line  ? 


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